UNIVERSITY  OF  CALIFORNIA. 

GIFT  OF 

PACIFIC  THEOLOGICAL  SEMINARY. 
Accession  Class 


ARITHMETIC, 

PRACTICALLY  APPLIED, 

FOR   ADVANCED   PUPILS, 
AND  FOR  PRIVATE  REFERENCE, 


DESIGNED   AS    A 


SEQUEL  TO  ANY  OF  THE  ORDINARY  TEXT-BOOKS  ON  THE  SUBJECT. 


BY    HORACE    MANN,    LL.D., 

THE    FIRST    SECRETARY    OF    THE    MASSACHUSETTS    BOARD    OF    EDUCATION, 

AND    PLINY    E.    CHASE,    A.M., 

AUTHOR    OF  'THE  COMMON-SCHOOL    ARITHMETIC.' 


R  A 

OF  THE 

C    UNIVERSITY   ) 


PHILADELPHIA: 
PUBLISHED  BY  E.  II.  BUTLER  &  CO. 

1857. 


VO 


Entered,  according  to  the  Act  of  Congress,  in  the  year  1850,  by 
HORACE  MANN  AND  PLINY  E.  CHASE 

in  the  Clerk's  Office  of  the  District  Court  of  the  United  States,  in  and  lor  the 
Eastern  District  of  Pennsylvania. 


JUST    PUBLISHED, 

THE   ELEMENTS  OF  ARITHIVIETIC ,  PART   FIRST,   for 

Primary  Schools.  The  SECOND  PART,  for  Grammar  Schools,  which 
is  in  course  of  preparation,  will  complete  the  following  Arithmetical 
Series : — 

ELE31ENTS  OF  ARITHMETIC,  PART  FIRST.    By  HORACE  MANN 

and  PLINY  E.  CHASE. 
ELEMENTS  OF  ARITHMETIC,  PART  SECOND.   By  HORACE  MANN 

and  PLINY  E.  CHASE. 

COMMON-SCHOOL  ARITHMETIC.    By  PLINY  E.  CHASE. 
ARITHMETIC   PRACTICALLY  APPLIED.      By  HORACE  MANN  and 

PLINY  E.  CHASE. 


E.  B.   HEARS,   6TEREOTYPER.  C.   SHERMAN,   PRINTER. 


MR.  MANN'S  PREFACE. 

THE  appearance  of  my  name  on  the  title  page  of  this  Arithme 
tic,  requires  me  to  state  the  extent  of  my  connection  with  its 
authorship,  and  of  my  responsibility  for  its  execution. 

Believing  the  idea  of  the  work  to  be  original,  I  will  attempt  its 
elucidation.  In  seeking  for  the  elements  or  materials  of  its  ques 
tions,  it  proposes  to  take  a  survey  of  all  the  vocations  of  life,  of  ! 
all  the  facts  of  knowledge,  and  of  all  the  truths  of  science,  and  to; 
make  a  selection  from  each  department  of  whatever  may  be  most 
interesting  and  valuable.  It  does  not  confine  itself  to  the  play 
things  of  the  nursery,  or  to  the  commodities  of  the  market  place, 
and  to  the  money  they  will  cost,  or  make,  or  lose.  On  the  contrary, 
the  present  work  proposes  to  carry  the  student  over  the  wide  ex 
panse  of  domestic  and  social  employments ;  to  introduce  him  to 
the  various  departments  of  human  knowledge  so  far  as  that  knowl 
edge  has  been  condensed  into  tables,  or  exhibited  in  arithmetical 
summaries,  and  to  make  him  acquainted  with  many  of  the  most 
wonderful  results  which  mathematical  science  has  revealed.  In 
stead  of  groping  along  the  mole-path  of  an  irksome  routine,  with 
little  other  change  than  from  dollars  and  cents,  to  pounds  and 
pence,  or  some  other  familiar  currency,  and  with  little  other  vari 
ety  than  from  cloth  to  corn,  or  some  other  common-place  commod 
ity,  it  derives  its  examples  from  biography,  geography,  chronol 
ogy,  and  history  ;  from  educational,  financial,  commercial,  and 
civil  statistics ;  from  the  laws  of  light  and  electricity,  of  sound 
and  motion,  of  chemistry  and  astronomy,  and  others  of  the  exact 
sciences.  Trades,  handicrafts,  and  whatever  pertains  to  the  use 
ful  arts,  so  far  as  they  are  the  subject  of  numerical  statement, 
and  their  facts  possess  arithmetical  relations,  together  with  all 
the  ascertained  and  determinate  results  of  economical  or  political 
knowledge,  and  of  scientific  discoveries,  are  laid  under  contribu 
tion,  and  are  made  to  supply  appropriate  elements  for  the  ques 
tions  on  which  the  youthful  learner  may  exercise  his  arithmetical 
faculties. 

(iiO 


iv  MR.  MANN'S  PREFACE. 

In  this  way,  and  without  departing  from  the  most  rigid  rules 
on  which  an  arithmetical  text-book  should  be  constructed,  I  have 
supposed  that  a  work  may  be  prepared  which  shall  exemplify  in 
the  best  manner  the  science  of  numbers,  and  be  full  of  useful 
knowledge  also;  and  which,  while  it  exercises  the  student's  powers 
of  calculation,  shall  enlarge  his  acquaintance  with  the  varied 
business  of  the  world,  and  with  many  of  the  most  interesting 
results  of  applied  science. 

In  a  universe  like  this,  where  every  star  has  been  weighed  in  a 
mathematical  balance,  and  all  inter-stellar  spaces  have  been 
measured  by  a  mathematical  line ;  where  the  orbits  of  all  the 
planets  have  been  traced  as  by  a  compass,  and  their  velocities 
graduated  to  their  distances  by  an  unchanging  law ;  where  not 
only  wind  and  tide,  but  every  particle  of  dust  in  a  hurricane,  and 
every  drop  of  water  in  a  cataract,  know  their  exact  places  by  an 
infallible  rule  ;  where  the  gravitation  of  matter,  the  radiation  of 
heat,  and  the  diffusion  of  light,  at  all  times  and  instantaneously, 
adjust  their  force  to  their  distance  with  unerring  precision ; 
where  every  chemical  combination  is  formed  on  some  fixed  prin 
ciple  of  proportion,  and  the  atoms  of  every  crystal  arrange 
themselves  around  their  nucleus  in  geometric  lines ;  and  where, 
whatever  other  contemplations  or  volitions  occupy  the  Infinite 
Mind,  it  is  still  true,  as  was  said  by  the  old  Greek  philosopher, 
that  "God  geometrizes ;" — I  say,  in  such  a  universe,  built, 
weighed,  measured,  compounded,  and  arranged  on  mathematical 
principles,  why  should  not  the  arithmetical  exercises  of  those 
minds  which  have  entered  it,  to  dwell  in  it  forever,  embrac-e 
something  more  than  the  market  price  of  commodities,  the  gain 
or  loss  in  trade,  and  the  interest  and  discount  of  banks  ? 

Why,  for  instance,  cannot  a  child  be  taught  to  count  the  bones 
in  his  hands,  as  well  as  the  nuts  in  his  pocket ;  to  add  together 
the  number  of  bones  in  all  the  different  parts  of  his  body,  or  to 
subtract  the  number  of  those  contained  in  his  head  or  in  his 
hands,  from  the  number  of  those  contained  in  his  feet,  as  well  as 
to  add  or  subtract  the  number  of  apples  or  of  cakes  in  the  pos 
session  of  James,  John,  and  Joe ;  and  why  can  he  not,  by  such 
exercises,  be  led  to  enrich  his  mind  with  anatomical  or  physio 
logical  facts,  instead  of  stimulating  his  imagination  with  the 
provocatives  of  appetite  ?  Why  cannot  a  child  add  together  the. 
population  of  the  different  States  of  this  Union,  or  of  the  different 
nations  of  Europe  or  of  the  world,  and  thus  learn  the  sum  of 
the  population  of  the  whole  earth  and  of  its  parts,  as  well  as  to 


MK.  MANN'S  PREFACE.  v 

add  naked  columns  of  abstract  figures  ?  In  addition,  subtraction, 
multiplication,  and  division,  why  cannot  the  pupil  use  examples 
whose  elements  or  data  are  drawn  from  the  distance  between 
different  historical  epochs,  from  the  ages  of  distinguished  men, 
from  the  date  of  one  discovery  or  invention  to  that  of  another 
discovery  or  invention,  or  from  the  rise  to  the  fall  of  dynasties, 
or  from  a  compnrison  of  the  heights  of  different  mountains,  or 
the  lengths  of  different  rivers,  or  of  degrees  of  longitude  on 
different  parallels  of  latitude,  or  the  distance  from  city  to  city 
by  land,  or  from  port  to  port  by  sea ;  and  thus  live  in  the  per 
petual  presence  and  company  of  most  important  truths  pertaining 
to  history,  chronology,  biography,  and  geography,  and  so  fa 
miliarize  himself  with  these  classes  of  facts,  without  devoting 
special  time  or  effort  to  their  acquisition,  just  as  he  becomes 
familiar  with  the  faces  and  the  names-  of  his  school-fellows  and 
his  townsmen,  merely  because  he  has  always  lived  amongst  them? 
If  the  arithmetical  exercises  of  the  pupil  direct  his  attention 
almost  exclusively  to  the  shop  of  the  retailer  or  the  counting- 
room  of  the  merchant,  then  he  does  not  enjoy  even  a  pedlar's 
opportunity  to  become  acquainted  with  men,  events,  times, 
places,  and  things, — with  the  great  results  of  business  and  of 
civilization,  as  they  now  exist  in  the  world.  Instead  of  wearying 
the  learner  with  endless  reiterations  about  bales,  boxes,  barrels, 
and  bushels,  or  dollars,  dimes,  cents,  and  mills,  why  not  open 
before  him  some  of  the  vast  storehouses  of  truth,  and  display 
some  specimens  of  their  endless  variety  and  beauty?  Let  the 
teacher,  taking  the  learner  by  the  hand,  follow  the  farmer,  the 
craftsman,  the  architect,  the  manufacturer,  the  road-maker,  the 
mill-wright,  the  ship-wright,  the  watch-maker,  and  the  long  cata 
logue  of  others  who  are  employed  in  the  mechanic  arts,  or  other 
branches  of  useful  industry;  or,  rising  into  the  sphere  of  the 
educated  or  professional  laborer,  let  him  observe  the  optician,  the 
electrician,  the  mathematical  instrument  maker,  the  astronomical 
observer,  the  telegraph  operator,  and  borrow  from  them  all  some 
of  the  curious  facts  pertaining  to  their  respective  arts  and  pro 
fessions,  and  convert  them  into  the  pleasing  and  instructive 
elements  of  arithmetical  problems.  I  can  conceive  of  a  work  so 
replete  with  the  facts  of  technology  and  science,  that  it  shall  be 
examined  with  interest  and  profit  by  any  one  who  is  only  seeking 
after  valuable  information,  and  which,  at  the  same  time,  shall 
be  perfectly  adapted  to  the  mere  student  who  only  seeks  after  the 
best  means  of  arithmetical  practice. 


vi  MR.  MANN'S  PREFACE. 

Among  the  advantages  of  such  a  work  as  is  here  proposed,  the 
two  following  seem  to  me  unquestionable: 

1 .  The  pupil,  while  studying  Arithmetic  for  its  own  sake,  will 
acquire  some  knowledge  of  many  other  things.     Cicero  observes 
that  the  face  of  a  man  will  be  tinged  by  the  sun,  for  whatever 
purpose  he  may  walk  abroad.     So,  by  daily  familiarity,  the  mind 
of  the  student  will  be  replenished  with  useful  facts,  and  imbued  with 
a  scientific  spirit,  although  the  acquisition  of  the  facts  and  the 
spirit  be  not  the  direct  object  of  his  study.     So  far  as  the  examples 
of  his  text-book  have  been  drawn  from  the  actual  business  of  life, 
as  it  goes  on  around  him,  the  learner  cannot  but  see  that  his 
studies  have  a  practical  bearing  and  are  connected  with  obvious 
realities.     A  great  number  of  facts,  such  as  dates,  sums,  quan 
tities,  distances,  will  be  impressed  on  his  mind  ;  and  thus  a  species 
of  most  valuable  knowledge,  which,   as  we  all  know,  it  is  most 
difficult  to  acquire  in  after-life,  will  be  gratuitously  bestowed  upon 
him.     Doubtless  great  differences  will  exist  among  pupils,  in  re 
gard  to  the  amount  of  information  they  will  obtain  from  being 
domiciled,  as  it  were,  among  the  determinate  truths  of  business, 
art  or  science ;  but  even  the  most  dull  and  stupid  will  be  con 
strained  to  learn  something,  not  only  of  the  existence  of  various 
sciences,  and  of  various  kinds  of  business,  of  which  they  would 
otherwise  be  forever  ignorant,  but  also  of  the  nature   and  dis 
tinctive  characteristics  of  those  sciences  or  employments.     All  will 
be  saved  from  great  misconceptions ;  and  doubtless  the  curiosity 
of  many  will  be  awakened  for  further  information.     These  advan 
tages  embrace  not  only  an  increase  of  positive  knowledge,  but  an 
enlargement  of  the  mind's  scope  in  regard  to   the  subjects  of 
knowledge. 

It  is  unnecessary  here  to  remind  the  observing  man,  how  the 
understanding  of  any  one  thing,  by  the  self-activity  of  the  faculties, 
generates  the  power  of  understanding  many  other  things,  and 
each  of  these,  in  their  turn,  of  many  more,  and  so  on  in  geometri 
cal  progression.  The  knowledge  of  any  one  truth  acts  as  an  intro 
ducer  and  interpreter  between  us  and  all  its  kindred  truths. 

2.  There  are  advantages  of  another  kind,  which  appear  to  me 
of  not  inferior  magnitude,  though  I  am  fully  aware  that  the  views  I 
am  about  to  present,  will  affect  different  minds  very  differently, 
according  to  their  Theory  of  Mind.    My  belief  is  that  Arithmetic, 
in  its  strict  and  technical  sense,  addresses  but  one  faculty  of  the 
mind,  or,  at  most,  but  a  very  limited  group  of  the  faculties.     No 


MR.    MANN  S   PREFACE.  Vll 

other  study  pursued  in  our  schools  is  so  restricted,  either  in  re 
gard  to  the  mental  powers,  which  it  calls  into  exercise,  or  the 
objects  which  it  brings  under  their  cognizance.  Hence,  during 
the  hours  devoted  to  Arithmetic  in  our  schools,  most  of  the  mental 
faculties  lie  dormant,  or  play  the  truant  by  employing  themselves 
upon  forbidden  objects.  Doubtless  this  intense  exercise  of  a  sin 
gle  faculty,  or  of  a  limited  group  of  faculties,  and  the  non-exercise 
of  all  the  rest,  is  one  of  the  main  reasons  why  Arithmetic,  when 
not  taught  with  great  ability,  is  so  often  an  irksome  study. 
Reading  and  Geography,  for  instance,  cover  the  widest  field  of 
interesting  subjects,  and  it  is  impossible  for  any  teaching  to  be  so 
dull,  or  any  circumstances  so  repulsive,  as  wholly  to  despoil  them 
of  their  charms.  If  we  would  invest  Arithmetic  with  similar  attrac-  / 
tions,  we  must  draw  its  examples  from  as  wide  and  as  rich  a  field/ 
Then  will  it  interest  new  faculties, — faculties  which,  otherwise,  U; 
never  addresses.  All  metaphysicians  know,  from  the  principles 
of  their  science,  and  all  laborious  students  know,  from  their  own 
experience,  that  nothing  refreshes  or  re-creates  a  wearied  faculty 
so  certainly  and  so  speedily  as  the  genial  exercise  of  some 
other  faculty.  Such  exercise  is  more  restorative  than  absolute 
quiescence.  It  is  with  the  faculties  of  the  mind  as  with  the 
muscles  of  the  body,  they  should  have  alternate  exercise  and  rest ; 
and  the  most  healthful  and  agreeable  relaxation  for  any  organ 
that  is  tired  of  exercise,  is  the  exercise  of  another  organ  that  is 
tired  of  repose.  The  footman  who  travels  over  a  long  and  level 
road,  where  the  same  muscles  are  subjected  to  a  perpetual  recur 
rence  of  the  same  strain,  rejoices  at  the  sight  of  a  hill  in  the  dis 
tance  ;  for  he  prefers  to  put  a  new  set  of  muscles  to  the  hard 
service  of  carrying  his  body  up  a  hill,  rather  than  to  compel  the 
fatigued  ones  to  continue  their  lighter  task.  Equally  cheering 
and  recuperative  must  the  alternation  be,  when,  after  addressing 
one  set  of  faculties  with  one  combination  of  agreeable  truths,  we 
appeal  to  another  set  of  faculties  to  determine  certain  arithmetical 
relations  which  exist  between  those  truths. 

Should  any  teacher  dissent  from  this  doctrine,  that  alternate 
exercise  and  rest,  like  the  ever-alternating  systole  and  diastole  of 
the  heart,  is  the  law  of  all  our  powers,  both  bodily  and  spir 
itual ;  or  should  any  pupil  be  found,  whose  senses  are  such  per 
fect  non-conductors  of  truth,  as  to  exclude  all  information  though 
the  atmosphere  by  which  he  is  surrounded  is  saturated  with  it, 
still,  no  loss  or  harm  would  be  incurred ;  for  the  single  process 


till  MR.    MANN  S   PREFACE. 

of  arithmetical  training  can  be  carried  on  by  the  aid  of  the 
examples  contained  in  this  book  as  well  as  by  those  in  any  other. 

Such  is  an  outline  of  the  present  work.  Since  its  conception 
first  flashed  upon  my  mind,  I  have  pondered  upon  it  much;  and 
have  conversed  respecting  it  with  many  gentlemen  not  only  of 
great  mathematical  attainments,  but  of  varied  scholarship,  and 
both  reflection  and  conversation  have  deepened  my  conviction  of 
its  value. 

Justice  to  my  associate,  Mr.  CHASE,  requires  me  to  say  a  word 
in  regard  to  the  shares  of  responsibility  and  of  merit  which  attach 
to  us  respectively  as  the  joint  authors  of  this  work.  In  commu 
nicating  my  plan  to  him,  I  unfolded  its  whole  scope  and  purpose 
as  it  lay  in  my  own  mind  ;  I  indicated  the  various  sources  whence 
materials  for  its  construction  might  be  drawn,  and  I  have  rendered 
him  some  aid  in  the  collection  of  those  materials.  The  residue  of 
the  work, — the  definitions,  the  rules,  the  statement  of  the  ques 
tions,  and  the  answers,  so  far  as  answers  are  given,  together  with 
the  arrangement  of  the  topics,  or  subjects, — is  substantially  his. 
So  far  as  there  is  skill  in  their  selection,  or  science  in  their  state 
ment,  or  accuracy  in  their  results,  I  shall  gladly  join  with  others 
in  awarding  the  merit  to  him. 

HORACE  MANN. 

MARCH,  1850. 


P.  E.  CHASE'S  PREFACE. 

THE  following  work  does  not  profess  to  be  a  mere  Arithmetic  of 
the  ordinary  stamp.  It  takes  ground  that  has  hitherto  been 
unocccupied,  and  its  plan  in  most  respects  is  entirely  new. 

In  the  "Common-School"  Arithmetic  the  rules  and  principles 
that  are  usually  taught,  have  been  very  fully  explained  and  illus 
trated,  but  in  that  work,  as  in  all  of  the  other  similar  treatises 
that  are  now  in  general  use,  the  primary  object  is  merely  the  incul 
cation  of  processes, — the  practical  application  of  those  processes 
being  introduced  only  incidentally.  But  every  teacher  is  aware 
that  practical  exercises,  more  numerous  than  it  would  be  possible 
to  insert  in  an  elementary  work,  without  rendering  it  of  an  incon 
venient  size,  are  necessary  in  order  to  make  a  thorough  arithme 
tician.  This  necessity  can  often  be  but  partially  supplied,  and 
consequently  most  pupils  find  whenever  they  enter  into  active  life, 
that  the  calculations  of  their  business,  whatever  that  business 
may  be,  are  all  to  be  learned  anew,  and  that  all  the  Arithmetic 
they  have  studied  at  school  is  of  little  value,  except  for  the  expert- 
ness  it  may  have  given  them  in  the  simple  operations  of  the  fun 
damental  rules. 

I  have  long  believed  that  a  Sequel  to  Arithmetic, — a  work  not 
designed  to  take  the  place  of  any  of  the  ordinary  text-books,  but 
absolutely  requiring  a  familiarity  with  some  one  or  more  of  them 
before  it  can  be  studied  at  all, — might  be  so  prepared  as  to  sup 
ply  the  want  to  which  I  have  alluded.  By  giving  numerous 
examples  similar  to  those  which  are  constantly  occurring  in  the 
various  walks  of  life,  the  student  may  be  enabled  to  prepare  him 
self  better  at  school,  for  his  future  employment,  and  by  the  inci 
dental  introduction  of  subjects  of  general  interest,  the  study  may 
be  made  pleasant,  and  the  thoroughness  which  is  of  the  first  im 
portance  in  every  undertaking  may  be  more  readily  secured. 

There  are  many  difficulties  connected  with  the  selection  and 
arrangement  of  materials  for  such  a  treatise,  which  may  be  urged 
in  excuse  for  any  deficiencies  that  exist  in  the  present  volume. 
I  trust,  however,  that  there  will  be  few  objections  brought 

(ix) 


X  PREFACE. 

against  the  execution  of  our  undertaking,  for  which  the  book 
itself  does  not  afford  a  remedy.  Any  fault  of  arrangement  may 
be  easily  corrected,  by  taking  up  the  chapters  in  a  different 
order  from  the  one  I  have  adopted ;  any  redundancy  of  exam 
ples  may  be  avoided,  by  omitting  such  portions  as  appear  of  the 
least  importance ;  any  deficiency  may  be  supplied,  by  framing 
additional  questions  from  the  abundant  materials,  in  the  shape 
of  tables  and  remarks,  which  are  interspersed  throughout  the 
book.  This  latter  peculiarity  of  our  plan,  we  regard  as  one  of 
its  greatest  recommendations.  Teachers  can  make  new  questions 
for  their  classes,  to  any  extent  they  deem  expedient,  and  they 
may  find  it  a  most  valuable  exercise  to  allow  their  pupils  to  form 
questions  for  each  other,  from  the  data  that  are  given  in  the 
text-book. 

In  many  places,  the  explanations  have  been  simplified  by  the 
use  of  letters.  If  their  use  should  seem,  at  first,  too  algebraical, 
I  think  all  objection  will  be  removed  upon  examination  of  the 
manner  in  which  they  are  employed.  Every  scholar  who  has  any 
tolerable  degree  of  familiarity  with  Arithmetic,  should  be  able  to 
reason  as  readily  with  as  and  bs,  as  with  apples  and  beans  ;  and 
I  flatter  myself  that  the  manner  in  which  I  have  applied  the 
simplest  rules  of  analysis  to  operations  with  letters  and  symbols, 
will  be  of  great  use  to  all  who  study  the  following  pages. 

The  character  of  the  work  having  rendered  it  necessary  that  it 
should  be,  like  other  school-books,  principally  a  compilation,  I 
have  endeavored  to  search  for  the  best  authorities,  and  to  give 
due  credit  to  the  various  writers  that  I  have  consulted.  I  am 
also  indebted  to  officers  of  the  government,  and  to  teachers  of 
some  of  our  leading  institutions  of  learning,  for  many  valuable 
suggestions.  I  have  added  many  rules  and  illustrations  that 
atfe  entirely  original,  and  I  hope  that  the  result  of  our  joint  labors 
will  be  found  a  valuable  companion  in  every  schoolroom  into 
which  it  may  be  introduced. 

I  consider  myself  most  fortunate  in  having  obta'  'ied  the  valuable 
assistance  of  my  associate,  HORACE  MANN.  His  ^roposal,  that  the 
work  should  be  rendered  highly  instructive  as  well  as  practical, 
will  doubtless  commend  itself  to  general  favor;  and  for  any  merit 
that  may  be  found  in  the  execution  of  the  plan  indicated  in  his 
preface,  I  am  greatly  indebted  to  the  suggestions  and  criticisms 
with  which  he  has  favored  me,  during  his  revision  of  the  manu 
script  and  proof-sheets.  PLINY  E.  CHASE. 


TABLE  OF   CONTENTS. 


ARTICLE  I. — SYMBOLS. 

Section  Page 

1.  Significations  of  Symbols »  15 

2.  Examples  for  the  Pupil 16 

ART.  II. — TEST  QUESTIONS. 

3.  Questions    on    the    Theory  of 

Arithmetic 17 

4.  Questions  on   the  Practice   of 

Arithmetic -. 25 

ART.   III.— THE  FUNDAMENTAL 
RULES. 

5.  Examples  in  Addition  and  Sub- 


traction. 


33 


6.  Examples  in  Addition,  Subtrac 

tion,  Multiplication,  and  Di 
vision  - 38 

7.  Table  for  Additional  Exercises  46 

8.  Fractions  and  Compound  Num 

bers 48 

9.  Table  of  Latitudes  and  Longi- 


tudes 


55 


10.  Miscellaneous  Examples 59 

11.  Test  Examples 64 

ART.  IV. — MEASURES,  WEIGHTS, 
AND  CURRENCIES. 

12.  Standards  of  the  United  States  66 

13.  Standards  of  Great  Britain 81 

14.  Standards  of  France 84 

15.  Miscellaneous  Table 87 

16.  Ancient     Measures,    Weights, 

and  Coins 104 

17.  Examples  for  the  Pupil 106 

ART.V. — THE  FARM. 

18.  Rules     for     determining      the 

Weight  of  Cattle 110 

Rules  for  Measu-ring  Grain 111 


[9 

20 

21.  Measurement  of  Land 113 

22.  Examples  for  the  Pupil 113 


Weight  of  Grain  and  Hay 112 


ART.  VI.— THE  GARDEN. 
23.  Examples  for  the  Pupil 


ART.  VII.—  THE  HOUSEHOLD. 

Section  Page 

24.  General  Information  ...........  128 

25.  Household  Mensuration  .......  129 

26.  Examples  for  the  Pupil  ........  131 

ART.  VIII.—  ARTIFICERS'  WORK. 

27.  The  Carpenter  and  Joiner  ......  136 

28.  The  Mason  ....................  MO 

29.  The  Bricklayer  ...............  142 

30.  The  Plasterer.  ...  .............  113 

31.  The  Painter  and  Glazier  .......  144 

32.  The  Paver,  Slater,  and  Tiler..  144 

33.  The  Plumber  ..................  145 

34.  Specification  and  Estimates  ____  146 

ART.  IX.  —  STRENGTH  OF  MATERIALS. 

35.  Flexibility  and  Strength  of  Tim 

ber  .........................  153 

36.  Problems   on   the   Strength    of 

Timber  ....................  153 

37.  Examples  for  the  Pupil  ........  156 

38.  Strength   of   Prisms    one   inch 

square  ......................  157 

39.  Lateral  Strength  of  Bars  ......  157 

40.  Cohesive  Force  of  Iron  ........  158 

41.  Resistance  to  Crushing  ........  158 

42.  Table  of  Cables  ..............  159 

43.  Weight  of  Stone  ..............  159 

44.  Problems   on    the   Strength   of 

160 
163 


Iron 
45.  Examples  for  the  Pupil 

ART.  X.—  SPECIFIC  GRAVITY. 


46.  Table  of  Specific  Gravities  ----  165 

47.  Problems  in  Specific  Gravity..  169 

48.  Examples  for  the  Pupil.  .  ..  .....  170 

ART.  XI.—  THE  ROAD. 

49.  General  Remarks  ........  .  .......  173 

50.  Examples  for  the  Pupil  ........  174 

ART.  XII.—  THE  ENGINEER. 


51.  The  Steam  Engine 178 

I  52.  The  Water  Wheel 180 

.123  1 53.  Pumps 183 


Xll 


TABLE    OF    CONTENTS. 


ART.  XIII.— THE  LABORATORY. 

Section  Page 

54.  Chemical  Combinations 184 

55.  Table  of  Chemical  Equivalents  186 

56.  Examples  for  the  Pupil 187 

ART.  XIV.— GENERAL  ANALYSIS. 

57.  Remarks    on    the    Solution   of 

Questions 189 

58.  Examples  illustrating  the  First 

Method 190 

59.  Examples   illustrating   the  Se 

cond  Method 195 

60.  Miscellaneous  Examples 196 

ART.  XV.— THE  COUNTING-HOUSE. 

61.  Percentage 208 

62.  Problems  in  Percentage 209 

63.  Examples  in  Percentage 211 

64.  Percentage    on    Sterling     Mo 

ney 213 

65.  Banking 215 

66.  Partial  Payments 218 

67.  Legal  Interest 219 

68.  Equation  of  Payments 223 

69.  Accounts  Current 226 

70.  Practice 229 

71.  Cause  and  Effect. 236  | 

72.  Exchange 242 

73.  Arbitration  of  Exchange 255  j 

74.  Alligation 258  • 

75.  General  Average 263  j 

ART.  XVI.— STATISTICS. 

76.  Product  of  Mines 265 

77.  Agricultural  Products 266 

78.  Agricultural    Products,    conti 

nued 267 

79.  Occupations 268 

80.  State  of  Education 269 

81.  Products  of  Industry 270 

ART.  XVII. — PERMUTATION  AND 
COMBINATION. 

62.  Permutation 271 

83.  Combination 273 


ART.  XVIII. — INVOLUTION  AND  EVO 
LUTION. 

Section  Page 

84 .  Involution 274 

85.  Evolution 276 

86.  Roots  of  All  Powers 278 

87.  Application  of  the  Square  Root  282 

88.  Application  of  the  Cube  Root2S4 

ART.  XIX. — PROGRESSION,  OR 
SERIES. 

89.  Arithmetical  and  Geometrical 

Progression 286 

90.  Harmonical  Progression 289 

91.  Compound  Interest ...291 

92.  Annuities 292 

ART.  XX.— POSITION. 

93.  Single  Position 300 

94.  Double  Position 301 

ART.  XXI. — APPROXIMATIONS. 

95.  Multiplication 305 

96.  Division 306 

97.  Continued  Fractions 307 

98.  Evolution 311 

99.  Examples  in  Approximation.  .313 

ART.  XXII. — PROPERTIES  OF  NUM- 


100.  Properties     of    Squares     and 

Cubes 313 

101 .  Prime  and  Composite  Numbers3l6 

102.  Figurate  Numbers 320 

103.  The  Fundamental  Rules 321 

104.  Curious  Problems 325 

ART.  XXIII.— MISCELLANEOUS  PROB 
LEMS. 

105.  Chronology 327 

106.  The  Moon's  Age  and  Southing  335 

107.  Mensuration 338 

108.  Tonnage  of  Vessels 342 

109.  Gauging 344 

110.  Miscellaneous  Examples 347 

111.  Table  of  Prime  and  Composite 

Numbers ...376 


LIST  OF  AUTHORITIES. 


Allen,  W.,  American  Biographical 
and  Historical  Dictionary. 

American  Almanac. 

American  Temperance  Society, 
Reports. 

Babbage,  C.,  Economy  of  Ma 
chinery  and  Manufactures. 

Bridgewater  Treatise. 

Bache,  A.  D.,  Reports  on  Weights, 
Measures  and  Balances. 

Barlow,  P.,  New  Mathematical 
Tables. 

Barnard, H., School  Architecture. 

Beckman,  J.,  History  of  Dis 
coveries  and  Inventions. 

Belknap,  J.,  American  Biography. 

Benjamin,  A.,  Practice  of  Archi 
tecture. 

Bigelow's  Technology. 

Biographia  Americana. 

Biographic  Portative  Universelle. 

Bonnycastle's  Arithmetic. 

Boston  Custom-House,  Table  of 
Currencies. 

Bowditch,  N.,  Practical  Navi 
gator. 

British  Husbandry. 

British  Sanitary  Reports. 

Budge,  J.,  Practical  Miner's 
Guide. 

Buist,  R.,  American  Flower  Gar 
den  Directory. 

Carpenter,  W.  B.,  Mechanical 
Philosophy,  Horology,  and 
Astronomy. 

Carpenters'  and  Builders'  As 
sistant. 

Cavallo,  T.,  Elements  of  Natural 
Philosophy. 


Chambers's  Educational  Course. 
Information  for  the  People. 


Colman,H., Agricultural  Reports. 

European  Agriculture. 

Crossley  &  Martin,  Intellectual 
Calculator. 

Daniell,  J.  F.,  Chemical  Phi 
losophy. 

Deane,  S.,  New  England  Farmer. 

Desaguliers,  J.  T.,  Course  of 
Experimental  Philosophy. 

Draper,  J.  W.,  Text  Book  on 
Chemistry. 

Emerson,  G.  B.,  Report  on  the 
Forests  of  Massachusetts. 

Encyclopaedia  Americana. 

Britannica. 

Edinburgh. 

Rees'. 


Evans,    0.,    Young   Mill-Wright 

and  Miller's  Guide. 
Ewing,    A.,    Practical     Mathe 
matics. 
Ferguson,  J.,  Lectures  on  Select 

Subjects. 
Gillespie,    W.    M.,    Manual    of 

Road  Making. 
Gregory,  G.,  Dictionary  of  Arts 

and  Sciences. 
Griffiths,  T.,    Chemistry  of  the 

Four  Seasons. 
Hare,  R.,  Chemistry. 
Hatfield,  R.  G.,  American  House 

Carpenter. 
Herschel,    Sir   J.,    Outlines    of 

Astronomy. 

Hunt,  F.,  Merchants'  Magazine. 
Hutton,C.,  Recreations  in  Mathe 
matics  and  Natural  Philosophy. 

(13) 


XIV 


LIST    OF    AUTHORITIES. 


Ingram,  A.,  Concise  System  of 
Mathematics. 

Jackson,  C.  T.,  Report  on  the 
Geology  and  Mineralogy  of 
New  Hampshire. 

Jamieson,  A.,  Mechanics  of 
Fluids. 

Johnson,  C.  W.,  Farmer's  Ency 
clopaedia. 

Johnson,  G.  W.,  Dictionary  of 
Modern  Gardening. 

Johnson,  L.  D. ,  Memoria  Technica. 

Johnson,  W.  R.,  Report  on 
American  Coals. 

Keith,  T.,  Complete  Practical 
Arithmetician. 

Lavoisne,  C.  V.,  Genealogical, 
Historical,  Chronological,  and 
Geographical  Atlas. 

Leslie,  J.,  Philosophy  of  Arith 
metic. 

Library  of  Useful  Knowledge. 

Liebig,  J.,  Animal  Chemistry. 

Lloyd,  H.,  Lectures  on  the  Wave 
Theory  of  Light. 

London,  Edinburgh,  and  Dublin 
Philosophical  Magazine. 

Loudon,  J.  C.,  Encyclopaedia  of 
Agriculture. 

M'Culloch,  J.  R.,  Commercial 
Dictionary. 

M'Gauley,  J.  W.,  Lectures  on 
Natural  Philosophy. 

Mahan,D.  H.,  Elementary  Course 
of  Civil  Engineering. 

Mass.  Board  of  Education,  An 
nual  Reports. 

Mint,  U.  S.,  Manual  of  Coins. 

Moseley,  H.,  Illustrations  of  Me 
chanics. 

Murray,  H.,  Encyclopaedia  of 
Geography. 

Nesbit,  A.,  Treatise  on  Practical 
Arithmetic . 


Newton,  I., Universal  Arithmetic. 

Nicholson,  J.,  Operative  Me 
chanic. 

Norie,  J.  W.,  Epitome  of  Prac 
tical  Navigation. 

Owen,  R.  D.,  Hints  on  Public 
Architecture. 

Parnell,  E.  A.,  Applied  Chemistry. 

Partington,  C.  F.,  Account  of 
the  Steam  Engine. 

Patent  Office  Reports. 

Pierce,  B.,  Elementary  Treatise 
on  Sound. 

Pilkington,  J.,  Artists'  Guide  and 
Mechanics'  Own  Book. 

Pratt,  J.  H.,  Mechanical  Phi 
losophy. 

Priestley,  J.,  Description  of  a 
System  of  Biography. 

Ranlett,  W.  H.,  The  Architect. 

Rogers,  J.,  Vegetable  Cultivator. 

Scientific  American. 

Shaw,  E.,  Practical  Masonry. 

Smeaton,  J.,  Experimental  In 
quiry  concerning  the  natural 
powers  of  water. 

Somerville,  M.,  Connexion  of  the 
Physical  Sciences. 

Tredgold,T.,  Principles  of  Warm 
ing  and  Ventilating  Public 
Buildings. 

Tucker,  G. , Progress  of  the  United 
States  in  fifty  years. 

United  States  Almanac. 

Ure,    A.,    Dictionary    of    Arts, 
Manufactures  and  Mines. 
The  Cotton  Manufacture 


of  Great  Britain. 

Wade,  J.,  British  History,  Chro 
nologically  Arranged. 

Walsh, M., Mercantile  Arithmetic. 

Wistar,  C.,  System  of  Anatomy. 

Year  Book  of  Facts. 


ARITHMETIC. 


I.  SYMBOLS. 

1 .  THE  following  characters,  or  symbols,  are  frequently 
employed  in  Arithmetic  to  represent  the  operations  that  are 
to  be  performed  upon  quantities : 

The  sign  +  (plus  or  and)  shows  that  the  numbers  be 
tween  which  it  is  placed,  are  to  be  added  together.  Ex. : 
5  +  6;  3  +  10  +  27. 

The  sign  —  (minus  or  less)  shows  that  the  quantity 
which  follows  it,  is  to  be  subtracted  from  the  one  which  pre 
cedes  it.  Ex.:  11—8;  29-16. 

The  sign  —  (equal')  shows  that  the  quantity  which  fol 
lows  it,  is  equivalent  to  the  quantity  which  precedes  it. 
Ex. :  4  +  19=27  —  18  +  14  (read  4:  plus  19  equal  27  minus 
18^wsl4);  65-27  +  3=41. 

The  sign  X  (times')  shows  that  the  numbers  between 
which  it  is  placed,  are  to  be  multiplied  together.  Ex.  : 
4x15  =  60;  3x10  =  6x5.  Multiplication  is  also  some 
times  expressed  by  a  point,  thus :  4-15  =  60;  3-10  =  6-5. 

The  sign  -h  or  :  (divided  by)  shows  that  the  former  of 
two  quantities  is  to  be  divided  by  the  latter.  Ex.  :  240 -f- 
15  =  16;  108  :  9  =  36  :  3.  Division  is  also  represented  by 
placing  the  dividend  above,  and  the  divisor  below  a  hori- 
sontalline.  Es. :  ^-95  +  !5  =  14. 

A  vinculum        ;  or  a  parenthesis  (   ),  shows  that  several 

(16) 


16  SYMBOLS.  [ART.  II. 

quantities  are  to  be  collected  into  one.    Ex.  :  (8—4  +  16-f-2) 
X3=36;  (2x7—  11)  X  3  =  14^11x3=3x3  =9. 

A  small  figure  placed  above  any  quantity,  at  the  right 
hand,  denotes  a  power  of  that  quantity.  Ex.:  92:=81; 
(1  +  3)3  =  64; 


The  radical  sign  v7,  prefixed  to  a  quantity,  is  used  to 
denote  some  root  of  that  quantity.  If  no  figure  is  written 
above  the  sign,  the  square  root  is  indicated.  To  express  the 
3d,  4th,  or  nth  root,  3,  4,  or  n,  is  placed  at  the  left  hand.  Ex  : 


=  V24—  8=V16=2. 
2.  EXAMPLES  TO  BE  READ  AND  SOLVED  BY  THE  PUPIL. 

1.  8  +  9  +  7-3  +  6-5-11=?  An*.  11. 

2.  4-2  +  6-1  +  13  xa5=? 

3.  8xa4—  3  +  5  +  16-27=? 

4.  27x3x5-^45=?  Am.  9. 

5.  (6x8)  +  (427l4x12°)-(65-:-5)=? 

6.  (2/X342x3:2)  +  (15"xl2-10)=?  Am.  206. 

7.  (4  x  11  x  3)-(6T3  X  144  :  36x  '536)=  ? 


9. 


22—  (37+3><5)— 


II.  TEST  QUESTIONS. 

EVERY  pupil  should  be  required  to  give  written  answers 
to  as  many  of  the  questions  in  this  article  as  he  is  able  to 

a  The  pupil  should  be  taught  to  say,  13  times  5,  8  times  4,  rather 
than  13  multiplied  by  5,  8  multiplied  by  4. 


§3.]  THEORY   OF   ARITHMETIC.  17 

solve.  The  teacher  will  thus  understand  the  proficiency  of 
each  member  of  the  class;  and  can  give  such  explanations 
as  may  be  most  desirable.  The  questions  should  be  reviewed 
from  time  to  time,  until  they  can  all  be  answered  without 
any  mistakes.1 

3-  QUESTIONS  ON  THE  THEORY  OF  ARITHMETIC. 

1.  What  is  ARITHMETIC  ?    What  is  a  number  ?    In  what  different 
ways  may  numbers  be  expressed  ?     What  are  the  digits,  and  why 
are  they  so  called  ?     What  different  names  has  the  character  0  ? 
Can  you  give  any  reason  for  calling  naught  the  figure  of  place  ? 
What  are  the  fundamental  operations  of  Arithmetic  ?     What  is 
given,  and  what  is  required  in  each  ? 

2.  What  is  NUMERATION  ?    What  method  of  numeration  is  gene 
rally  adopted  ?     What  is  the  peculiarity  of  that  method  ?     What 
is  a  unit  ?     What  is  meant  by  the  simple  value  of  a  figure  ? — by 
the  local  value  ?     What  is  the  difference  between  a  place  and  a 
period?     What  names  are  given  to  the  places  in  each  period  ? 
How  do  you  read  units  ? — tens  ? — tens  and  units  ? — hundreds  ? — 
hundreds,  tens,  and  units  ?     Kepeat  the  names  of  the  first  eight 
periods.13     How  do  we   read  whole  numbers  ?     Can   the  Arabic 
method  represent  numbers  less  than  a  unit  ?    What  are  such  num 
bers  called  ?     How  are  they  distinguished  from  whole  numbers  ? 
How  do  we  read  decimals  ? 


a  "  A  teacher  ought  not  only  to  instruct  his  pupils,  but  also  to  inter 
rogate  them  frequently,  and  test  their  proficiency."  Quintilian. 

b  The  Numeration  table  may  be  continued  to  any  extent  we  please. 
The  periods  above  Sextillions,  to  Vigintillions,  are,  Septillions,  Octil 
lions,  Nonillions,  J)ecillions,  Undecillions,  Duodecillions,  Tredecil- 
lions,  QuatuordeciUions,  Quindecillions,  Sexdecillions,  Septendecil- 
lions,  Octodecillions,  Noverndecillions,  Vigintillions.  The  following 
illustrations  will  show  how  difficult  it  is  for  us  to  form  any  distinct  idea 
of  the  value  of  very  large  numbers. 

If  every  man,  woman,  and  child,  on  the  face  of  the  globe,  were  to 
count  at  the  rate  of  four  every  second,  without  ceasing  day  or  night, 
the  whole  amount  of  all  that  they  could  count  in  5000  years,  would  be 
less  than  one  sextillion. 

If  the  sun,  all  the  planets,  and  all  the  fixed  stars  that  are  visible 
through  the  most  powerful  telescope,  were  reduced  to  a  powder  so  fine 
that  100000  grains  would  be  less  in  bulk  than  a  drop  of  water,  and  if  one 
of  these  grains  were  destroyed  in  every  million  years,  the  whole  visible 
universe  would  probably  be  annihilated  in  less  than  one  vigintillion  years. 

2 


18  TEST   QUESTIONS   ON   THE  [ART.  II. 

3.  What  is  the  effect  of  removing  a  figure  two  places  to  the 
left  ? — two  places  to  the  right  ? — one  period  to  the  left  ? — to  the 
right  ? — two  periods  to  the  left  ? — to  the  right  ?     What  is  meant 
by  Notation  ?     How  do  you  write  whole  numbers  ?     In  writing 
five  sextillion,  thirty  billion,  and  seven  thousand,  what  would  you 
place  in  each  period  ?     If  you  were  to  omit  the  naughts  in  any 
one  period,  what  effect  would  it  have  on  all  the  figures  above  it  ? 
How  do  you  write  decimals  ?    How  would  you  make  5702  represent 
hundredths  ? — millionths  ? — thousands  ? — thousandths  ?    What  is 
the  effect  of  placing  naughts  at  the  right  hand  of  decimals  ? — at 
the   left   hand? — at   the   right   hand   of  integers?3 — at  the  left 
hand?     What  is  the  value  of  tens  multiplied  by  thousands? — 
hundreds  by  hundreds  ? — ten-thousands  by  tens  ? — hundred-thou 
sands  by  hundreds  ? — tens  by  hundreds  ? — thousands  by  thousands  ? 
— tenths  by  tens  ? — thousandths  by  hundreds  ? — thousandths  by 
hundredths  ? — thousands  by  hundredths  ? — millions  by  millionths  ? 
— How  would  you  represent  9000  by  adopting  9  for  the  base  of 
the  numerical  system? — by  adopting  5  for  the  base? 

4.  What  is  FEDERAL  MONEY  ?    How  may  it  be  written  ?    What  is 
the  probable  origin  of  the  sign  that  is  usually  prefixed  to  dollars  ? 
How  may  dollars  be  reduced  to  cents? — to   mills?     How  may 
cents  be  reduced  to  dollars? — to  mills  ?    How  may  mills  be  reduced 
to  cents  ? — to  dollars  ?     If  there  are  decimals  below  mills,  how 
may  they  be  read  ? 

5.  Since  removing  the  decimal  point  in  any  number  changes  the 
place  of  each  figure,  what  is  the  effect  of  removing  the  decimal 
point  two  places  to  the  left  ? — to  the  right  ? — three  places  to  the 
left? — to   the   right? — seven   places  to  the  left? — to  the  right? 
Then  how  can  you  most  readily  find  10,  100,  or  1000  times  any 
number? — .1,  .01,  .001,  or  .0001  of  any  numbe^? 

6.  What  is  ADDITIOX  ?     What  sign  is  used  to  denote  addition  ? 
What  is  the  sign  of  equality?     How  do  you  arrange  numbers  that 
are  to  be  added  together  ?     Where  do  you  commence  the  addition  ? 
What  do  you  do  with  the  sum  of  each  column?     Why  do  you  not 
always  place  the  whole  sum  of  a  column  underneath  the  column? 
How  do  you  find  whether  your  answer  is  correct  ?    Can  any  one  of 
a  series  of  numbers  be  greater  than  the  sum  of  the  whole  series  ? 
How  can  you  prove  addition  by  casting  out  9s  ? 

7.  What  is  SUBTRACTION  ?     What  is  the  remainder  ? — the  sub 
trahend? — the  minuend?    What  is  the  signification  of  the  termi- 

a  An  integer  is  a  whole  number  :  as  seven,  forty-nine. 


§3.]  THEORY   OF   ARITHMETIC.  19 

nation  nd  in  many  arithmetical  terms  ?     If  you  add  the  difference 
of  two  numbers  to  the  less  number,  what  will  you  obtain  ?     If 
you  take  the  difference  from  the  greater  number,  what  will  you 
obtain?     How  do  you  find  the  minuend,   if  the  remainder  and 
subtrahend  are   given? — the  subtrahend,   if  the  remainder  and 
minuend  are  given  ?     What  is  the  sign  of  subtraction  ?     Of  what 
operation  is  subtraction  the  opposite  ?    How  do  you  write  numbers 
in  subtraction  ?     Where  do  you  begin  to  subtract  ?     If  any  figure 
is  greater  than  the  one  above  it,  what  may  be  done  ?     Explain 
the  reason  of  this.     What  may  be  done  when  the  subtrahend  has 
more  decimal  places  than  the  minuend  ?     How  do  you  prove  sub 
traction  ?     In  finding  the  difference  between  two  numbers,  which 
must  be  the  minuend  ?     How  can  you  tell  which  is  the  larger 
number  ?     Can  you  think  of  any  method  of  proving  addition  by 
subtraction  ?     How  can  you  prove  subtraction  by  casting  out  9s  ? 
8.  What  is  MULTIPLICATION  ?     Of  what  is  it  an  abbreviation  ? 
What  is  meant  by  the  multiplicand  ? — the  product  ? — the  multi 
plier  ? — the  factors  ? — a  composite  number  ?     Name  some  compo 
site  numbers.      What   is  the    sign  of  multiplication?     Can  the 
multiplier  and  multiplicand  exchange  places  ?    Give  an  illustration 
in  proof  of  your  answer.     How  do  you  multiply  by  a  single  figure  ? 
How  may  you  multiply  by  a  composite  number  ?     How  do  you 
proceed  when  the  multiplier    consists  of   a  number  of  figures? 
How  many  decimals  must  there  be  in  the  product  ?     What  do  you 
do  if  there  are  not  decimals  enough?     What  may  be   done,  if 
there  are  naughts  at  the  right  hand  of  either  factor  ?     How  can 
you  most  readily  multiply  by  10,  100,  1000,  &c.  ?     Explain  the 
reason  for   the    several   modes  of  multiplication.     How  do  you 
prove  multiplication  ?     What  must  you  know  before  you  can  find 
the  value  of  any  number  of  things  ?     How  can  you  prove  multi 
plication  by  casting  out  9s  ?  • 

9.  What  is  DIVISION  ?  Of  what  is  it  an  abbreviation  ?  What 
is  meant  by  the  dividend  ?— the  quotient ?— the  divisor?— the 
remainder  ?  Of  what  are  the  divisor  and  quotient  factors  ?  What 
are  the  different  modes  of  expressing  division  ?  Give,  in  your 
own  words,  a  rule  for  division,  and  explain  the  reason  for  each 
step  of  the  process.  Of  what  operation  is  division  the  opposite  ? 
How  many  decimals  must  the  quotient  contain  ?  Why  ?  How  do 
you  find  the  dividend,  when  the  divisor  and  quotient  are  given  ? 
— the  divisor,  when  the  dividend  and  quotient  are  given  ?  What 
is  the  difference  between  short  and  long  division  ?  What  may  be 
done,  if  there  are  naughts  at  the  right  hand  of  the  divisor  ?  How 


20  TEST   QUESTIONS   ON   THE  [ART.  II. 

can  you  most  readily  divide  by  10,  100,  1000,  &c.  ?  How  do  you 
prove  division  ?  How  can  you  avoid  frequent  trials,  in  finding  the 
true  quotient  figure  ?  When  the  value  of  any  number  of  things 
is  given,  how  do  you  find  the  value  of  one  ?  When  the  value 
of  a  number  of  things  is  given,  how  can  you  find  the  number 
of  things  ?  Can  you  think  of  any  method  for  proving  multipli 
cation  by  division  ?  How  can  you  prove  division  by  casting  out  9s  ? 

10.  When  the  SUM  OF  Two  NUMBERS  and  one  of  the  numbers  are 
given,  how  do  you  find  the  other  ?    When  the  greater  of  two  num 
bers  and  their  difference  are  given,  how  do  you  find  the  less  ? 
When  the  less  of  two  numbers  and  their  difference  are  given,  how 
do  you  find  the  greater  ?     When  the  factors  are  given,  how  do 
you  find  the  product?     When  the  product  and  one  of  the  factors 
are  given,  how  do  you  find  the  other  factor?     When  the  value  of 
any  number  of  things  is  given,  how  do  you  find  the  value  of  any 
other  number  ?    By  which  of  the  fundamental  rules  do  you  increase 
a  number?    By  which  do  you  diminish  a  number?    In  what  cases 
would   you    employ  addition? — subtraction? — multiplication? — 
division?     What   is   meant   by  the  3d, — 7th, — 15th  power  of  a 
number  ? 

11.  Wrhat  is  a  PRIME  NUMBER  ? — a  composite  number  ?    Name  all 
the  prime  numbers  less  than  100.     How  can  you  tell  whether  any 
number  can  be  divided  by  2,  5,  3,  9  ?     How  do  you  find  the  prime 
factors  of  any  number?     How  do  you  determine  whether  any 
number  is  a  prime  ?     What  is  meant  by  a  multiple  ? — a  common 
multiple  ? — the  least  common  multiple  ?     How  do  we  find  the  least 
common  multiple  of  two  or  more  numbers  ?     Show  that  this  pro 
cess  will  give  the  least  common  multiple.     What  is  a  common 
divisor? — the    greatest   common  divisor?     How  do  we   find  the 
greatest  common  divisor  ?     Why  do  we  proceed  in  this  manner  ? 
What  is  cancelling  ?     How  do  you  cancel  ?     When  you  have  can 
celled  all  the  numbers  in  either  the  dividend  or  divisor,  what  must 
be  put  in  their  place  ?     When  the  divisor  is  a  composite  number, 
how  may  we  obtain  the  quotient? 

12.  What  are  FRACTIONS  ?     In  how  many  different  ways  may 
they  be  read  ?     Show  why  each  of  these  ways  may  be  adopted. 
What  is  meant  by  the  numerator  ? — the  denominator  ? — the  terms 
of  a  fraction  ?    WThat  is  shown  by  the  numerator  ? — by  the  denomi 
nator  ?     How  is  the  value  of  a  fraction  affected  by  increasing  the 
denominator  ? — the  numerator  ? — by  diminishing  the  numerator  ? 
— the  denominator  ?     What  is  a  mixed  number  ?     How  would  you 
find  what  part  324  is  of  187  ?     Why  ? 


§3.]  THEORY   OF   ARITHMETIC.  21 

13.  What  is  SEDUCTION  ?     How  do  we  reduce  fractions  to  whole 
numbers? — to   mixed   numbers? — to   decimals?     Explain    these 
reductions.     What  are   proper   fractions? — improper  fractions? 
When  is  a  fraction  greater  than  1  ? — less  than  1  ? — equal  to  one  ? 
How  would  you  write  a  decimal  in  the  form  of  a  fraction  ?    Explain 
the  rule  for  reducing  whole  or  mixed  numbers,  or  decimals,  to 
fractions.     Explain  the  method  of  reducing  compound  to  simple 
fractions ;  fractions  to  their  lowest  terms ;  fractions  to  a  common 
denominator ;  complex  to  simple  fractions ;    fractions  to  others 
having  any  given  numerator  or  denominator.     Explain  the  rules 
for  addition,  subtraction,  multiplication,  and  division  of  fractions. 
What  is  meant  by  an  integer  ? — by  the  reciprocal  of  a  number  ? 
Take  examples  of  your  own,  and  form  rules  for  finding  the  sum 
and  the  difference  of  two  fractions,  each  of  which  has  1  for  its 
numerator ; — for  finding  the  sum  and  the  difference  of  two  frac 
tions  which  have  the  same  numerator  ? 

14.  What  are  INFINITE  DECIMALS  ?    What  is  the  repetend  ?    How 
is  it  distinguished  ?    To  what  is  every  repetend  equivalent  ?    Show 
that  it  is  so.     How  then  would  you  reduce  any  infinite  decimal  to 
a  fraction  ?     How  may  addition  and  subtraction  of  infinite  deci 
mals  be  performed  without  reducing  them  to  fractions  ? 

15.  What  are  COMPOUND  NUMBERS  ?    Repeat  the  table  of  Federal 
Money  ;  English    Money  ;    Troy  Weight ;  Apothecaries'  Weight ; 
Avoirdupois  Weight ;    Long    Measure ;    Cloth    Measure ;    Square 
Measure  ;  Cubic  Measure  ;  Dry  Measure  ;  Liquid  Measure  ;  Cir 
cular  Measure  ;  Time  Measure.     How  many  shillings  in  a  guinea  ? 
How  many  dollars  in  a  pound  sterling?     How  do  we  find  the 
area*  of  a  rectangular  surface  ?     How  many  cubic  feet  make  a 
foot  of  wood,  or  a  cord  foot  ?     How  many  cord  feet  in  a  cord  ? 
How  many  cubic  feet  in  a  ton  of  round  timber  ? — a  ton  of  square 
timber  ? — a  ton  of  shipping  or  storage  ?    How  do  we  find  the  solid 
contents  of  any  rectangular  solid  ? 

16.  For  what  purpose  are  each  of  the  weights  and  measures 
used?     How  many  cubic  inches  in  a  half  peck? — in  a  common 
gallon  ? — in  a  gallon  of  milk  or  of  malt  liquor  ?     For  what  pur 
pose  is  the  hogshead  (measure)  used  ?     How  many  degrees  in  a 
quadrant? — in  a  sign  of  the  zodiac?     How  many  days  in  a  leap 
year  ?     What  years  are  leap  years  ?     Repeat  the  number  of  days 
in  each  month.     In  mercantile  business  how  many  days  are  con 
sidered  as  making  a  month  ? 

a  The  number  of  square  feet,  rods,  or  acres,  &c.,  in  any  surface,  is 
called  its  area. 


22  TEST    QUESTIONS    ON   THE  [ART.  II. 

17.  Give,  in  your  own  language,   a  rule  for  reducing  higher 
denominations  to  lower ;  lower  denominations  to  higher  ;  for  com 
pound  addition ;    compound  subtraction ;    compound  multiplica 
tion  ;  compound  division.     Explain  each  rule.    Can  fractions  and 
decimals  be  reduced  by  the  same  rules  as  whole  numbers  ? 

18.  What  is  meant  by  PER  CENT.  ?    How  do  we  compute  any  re 
quired    percentage  ?     Why  ?      What   is   meant   by   commission  ? 
—  insurance  ?  —  premium  ?  —  policy  ? — underwriters  ?  —  taxes  ?  — 
stocks? — the  par  value? — the  dividend  on  a  stock?     When  is  a 
stock  said  to  be  above  par? — below  par  ? — at  a  discount? — at  an 
advance?     What  part  of  the  original  cost  is  stock  worth,  that 
sells  at  16f  per  cent,  below  par  ? — at  12 £  per  cent,  advance  ? — at 
a  discount  of  17^  per  cent.  ? — at  25  per  cent,  above  par?     What 
is  meant  by  duties? — specific  duties  ? — ad  valorem  duties  ? — draft? 
— tare  ? — gross  weight  ? — net  weight  ?     In  custom-house  business, 
what  allowance  is  made  for  leakage  and  breakage? — for  draft? 
How  do  we  find  what  per  cent,  is  gained  or  lost  by  any  transaction  ? 

19.  What  is  INTEREST  ? — simple  interest? — compound  interest  ? 
What  is  the  principal  ? — the  rate? — the  amount?     Give,  in  your 
own  language,  the  General  Rule  for  Interest ; — the    Bank  and 
Business  Rule.     Show  how  each   rule  is  obtained.     What  is  a 
promissory  note  ? — an  endorsement  ?     What  is  the  usual  mode  of 
computing  interest  on  notes  that  are  settled  within   a  year  from 
their  date  ?    What  is  the  Legal  Rule  ?    How  do  we  find  the  amount 
of  any  sum  at  compound  interest  ?     How  do  we  find  the  compound 
interest  ? 

20.  How  do  you  find  THE  RATE,  when  the  principal,  interest,  and 
time  are  given  ? — the  time,  when  the  principal,  interest,  and  rate 
are  given  ? — the  principal,  when  the  time,  rate,  and  interest  are 
given  ? — the  principal,  when  the  time,  rate,  and  amount  are  given? 
Explain  the  reasons  for  each  of  these  methods. 

21.  What  is  DISCOUNT? — equitable  discount? — bank  discount? 
How  do  you  find  equitable  discount  ? — bank  discount  ?     What  are 
days  of  grace?     How  many  days  of  grace  are  usually  allowed? 

22.  What  is  ANALYSIS  ?     What  is  meant  by  known  and  unknown 
quantities  ?     What  should  we  endeavor  to  do,  in  solving  difficult 
questions  ?    How  may  this  usually  be  done  ?    Give  a  General  Rule 
for  analysis. 

23.  What  is  RATIO  ?    How  is  it  expressed  ?    How  may  it  be  indi 
cated  ?     Which  is  the  usual  mode,  and  how  is  it  read  ?     What  is 
the  antecedent? — the  consequent?     How  do  you  find  the  ratio 
between  different  denominations  ? 


§3.]  THEORY    OF   ARITHMETIC.  23 

24.  What  do  you  understand  by  REDUCTION  or  CURRENCIES  ?    Do 
the  people  of  the  United  States  ever  have  occasion  to  make  such 
a  reduction  ?    Why  ?    Name  the  currencies  in  common  use.     How 
may  each   be  reduced  to  Federal    Money?     How  may  Federal 
Money  be  reduced  to  each  currency  ?    How  do  you  reduce  English 
Money  to  Federal  Money,  including  the  premium  of  exchange  ? 
Federal  Money  to  English  Money  ?     Give  your  reasons.     What  is 
the  cause  of  the  premium  ? 

25.  What  do  you  understand  by  PRACTICE  ?   Repeat  the  table  of 
parts  of  a  dollar.     Give  some  examples  illustrating  the  use  of 
the  table.    What  is  meant  by  given  terms  ? — by  terms  of  demand  ? 
Give  the  rule  for  complex  analysis,  and  illustrate  it  by  an  example 
of  your  own. 

26.  What  are  DUODECIMALS  ?    In  what  are  they  used  ?    What  are 
the  denominations  of  duodecimals  ?    How  are  they  marked  ?    What 
are  those  marks  called  ?    How  do  you  add  or  subtract  duodecimals  ? 
How  do  you  find  the  product  of  any  two  denominations  ?     Why  ? 
Multiply  4  7'  10"  by  3"  5'".     How  do  you  find  the  quotient  of  any 
two  denominations?     Why?     Divide  8'  2"  6'"  10""  4'""  by  2'  8"  4'". 

27.  What  is  a  PROPORTION  ?    How  is  it  usually  written,  and  how 
read  ?    In  what  other  ways  may  it  be  written  ?    Why  ?    What  are 
the  extremes  of  a  proportion  ?— the  means  ?     Prove  that  the  pro 
duct  of  the  extremes  is  equal  to  the  product  of  the  means.     If 
any  three  terms  of  a  proportion  are  given,  how  may  the  other  be 
found  ?    Which  term  usually  represents  the  unknown  term  ?    How 
do  you  find  the  fourth  term  ?     What  class  of  questions  may  be 
stated  in  the  form  of  a  proportion  ?     How  would  you  arrange  the 
first  and  second  terms,  if  you  wished  the  fourth  term  to  represent 
the  multiplication  of  the  third  term  by  a  ratio  ? — the  division  of 
the  third  term  by  a  ratio  ?     What  do  you  understand  by  direct, 
and  by  inverse  ratio  ? 

28.  What  is  placed  as  the  THIRD  TERM  in  a  proportion  ?    State  the 
rule  for  simple  proportion.     What  is  this  rule  sometimes  called  ? 
Is  the  value  of  the  third  term  ever  affected  by  more  than  one 
ratio?     Illustrate   your  answer  by  an  example,  and  state  that 
example  in  the  form  of  a  compound  proportion.     Why  do  you 
invert  the  several  ratios  by  which  the  third  term  is  to  be  multi 
plied  ?     State  the  rule  for  compound  proportion. 

29.  What  is  the  origin  of  ARBITRATION  or  EXCHANGE  ?    Give  an 
example.     In  what  different  ways  may  arbitration  be  effected  ? 
Which  is  the  usual  method  ?     Repeat  the  Chain  Rule. 


24  TEST    QUESTIONS    ON    THE  [ART.  II. 

30.  What  is  FELLOWSHIP  ?    How  inay  it  be  solved  by  Proportion  ? 
What  is  Compound  Fellowship  ?    Give  the  rule.    Explain  examples 
of  your  own,  both  by  analysis  and  by  proportion,  in  Simple  and 
Compound  Fellowship. 

31.  What  is  meant  by  AVERAGE  ?    Give  an  example.    How  do  you 
find  the  average  of  a  series  of  quantities?     What  is  Alligation? 
— alligation  alternate  ?     What  is  the  rule  for  alligation  medial  ? 
What  is  Equation  of  Payments  ?     How  does  it  differ  from  alli 
gation   medial?     Give    the   rule.     On  what   supposition   is   this 
rule  founded  ?     What  is  done  with  fractions  of  a  day  ?     Give  the 
rule  for  alligation  alternate.     Explain  the  rule  by  an  example  of 
your  own. 

32.  What  is  INVOLUTION?    What  is  the  first  power  of  a  number? 
— the   second  power  ? — the  fifth   power  ? — the  eleventh   power  ? 
Wliat  is  the  exponent?     What  is  the  second  power  often  called, 
and  why  ? — the  third  power  ? 

33.  What   is  EVOLUTION?     What   is  the  root  of   a  number? 
What  is  the  radical  sign,  and  how  is  it  used  ?     What  is  the  square 
root  ?     How  may  we  determine  the  number  of  figures  that  the 
square   root  will  contain  ?     Explain  the   method   by  which  the 
square  root  is  found,  by  multiplying  39  by  39  and  extracting  the 
square  root  of  the  product.     What  must  be  done  when  any  trial 
divisor   is   not   contained   in   the  dividend?  —  when    any  figure 
obtained  for  the  root  proves  too  large  ?     How  may  approximate 
roots  be  obtained  ? 

34.  What  is  the  CUBE  ROOT  of  a  number  ?     How  may  we  deter 
mine  the   number    of  figures  that  the  cube  root  will  contain  ? 
Explain  the  method  by  which  the  cube  root  is  found,  by  raising 
39  to  the  third  power,  and  extracting  the  cube  root  of  the  result. 
What  must  be  done  when  the  trial  divisor  is  not  contained  in  the 
hundreds  of  the  dividend  ? — when    any  figure  obtained  for  the 
root  is  too  large  ?     How  may  approximate  roots  be  obtained  ? 
How  may  the  trial  divisors,  after  the  first,  be  conveniently  found  ? 

35.  What   is    a  SERIES  ? — a  natural  series  ? — an    arithmetical 
series? — a  geometrical  series? — a  harmonical  series?     What  is 
meant  by  Arithmetical  Progression  ?     What  are  the  extremes  ? — 
the  means  ? — the  common  difference  ?     Explain  the  rule  for  find 
ing  one  of  the  extremes  and  the  sum  of  all  the  terms ;  for  finding 
the  common  difference  and  sum  of  the  terms  ;  the  number  of  terms 
and   the    sum  of  the  terms ;  the  number  of  terms  and  common 
difference  ;  one  extreme  and  the  common  difference. 

36.  What  is  meant  by  GEOMETRICAL  PROGRESSION  ? — by  the  ratio  ? 


§4.]  PRACTICE   OF   ARITHMETIC.  25 

— the  extremes  ? — the  means  ?  Explain  the  rule  for  finding  the 
last  term,  when  the  first  term,  ratio,  and  number  of  terms  are 
given ;  for  finding  the  sum  of  all  the  terms.  How  may  compound 
interest  be  found  by  series  ? — by  the  table  ?  How  may  we  find  by 
the  table,  the  present  worth  of  any  sum  at  compound  interest  ? — 
the  time  in  which  any  principal  will  amount  to  a  given  sum  ? — 
the  rate  at  which  the  principal  will  amount  to  a  given  sum  in  a 
given  time  ? 

37.  What  is  an  ANNUITY  ? — an  annuity  certain  ? — a  contingent 
annuity  ? — a  perpetual  annuity  ? — an  annuity  in  possession  ? — an 
annuity  in  reversion  ? — the  present  worth  of  an  annuity  ?  How 
do  you  find  the  amount  due  on  an  annuity  ? — the  present  worth  of 
an  annuity  certain  ? — the  present  worth  of  a  perpetual  annuity  ? 
— the  present  worth  of  an  annuity  in  reversion  ? — an  annuity,  the 
present  worth  being  given  ?  What  does  permutation  show  ?  Ex 
plain  the  rule. 

4.  PRACTICAL  QUESTIONS. 

All  the  information  necessary  to  enable  the  pupil  to  answer 
these  questions,  will  be  found  in  the  body  of  the  work. 

1.  How  do  you  find  the  price  at  which  goods  must  be  sold,  in 
order  to  gain  any  required  amount  ?  2.  How  do  you  find  the 
original  cost,  the  selling  price  and  the  loss  being  known  ?  3.  How 
do  you  find  the  whole  amount  of  an  invoice,  from  the  original 
cost  and  the  charges  upon  the  merchandise  ?  4.  How  do  you  find 
the  amount  gained  by  .any  sale  ?  5.  The  loss  upon  a  sale  ?  6.  The 
net  proceeds  of  a  sale?  7.  The  balance  of  an  account?  8.  The 
average  price  of  several  ingredients?  9.  The  quantity  of  several 
ingredients,  that  will  make  a  mixture  of  a  given  average  value  ? 

10.  How  would  you  determine  the  number  of  feet  of  boards1 
in  the  floor  of  a  room?  11.  The  number  of  bricks  in  the  walls 
of  a  house?  12.  The  number  of  shingles  or  slatesb  on  a  roof? 
13.  The  number  of  clapboards0  on  a  house?  14.  The  amount  of 
painting  and  plasteringd  in  a  house  ?  15.  The  quantity  of  timber 
in  the  frame  of  a  house?  16.  The  cost  of  glazing6  a  house  ? 

a  Lumber  is  measured  by  Board  Measure,  1  foot  =  y1o  cub.  foot. 
b  1000  shingles,  or  500  slates,  are  allowed  to  100  square  feet. 
c  Give  the  data  requisite  to  determine  the  number. 
d  Painting  and  plastering  arc  estimated  by  the  square  yard. 
e  Glazing  is  estimated  by  the  square  foot.     Builders  usually  furnish 
windows  by  the  piece. 


26  TEST    QUESTIONS    ON    THE  [ART.  II. 

17.  How  would  you  find  the  amount  of  earth  to  be  removed  in 
excavating  a  cellar,1  when  the  surface  of  the  ground  is  level  ? 
18.  When  the  surface  is  uneven?13  19.  The  quantity  of  gravel0 
required  for  making  a  road?  20.  For  an  embankment?  21.  For 
an  embankment  on  uneven  ground,  the  top  of  the  bank  being  level  ? 
22.  The  quantity  of  stone  in  a  wall  ?d  23.  In  estimating  walls, 
how  would  you  allow  for  corners  ?  24.  For  windows,  doors,  gates, 
or  other  openings? 

25.  How  would  you  estimate  the  number  of  yards  of  carpeting 
required  for  covering  a  floor  ?  26.  The  number  of  rolls  of  paper e 
necessary  for  papering  a  room  ?  27.  The  quantity  of  canvass  in 
the  sails  of  a  ship  ?  28.  The  weight  of  a  stone  wall  or  pillar  ? f 
29.  The  weight  of  iron  pipes  or  pillars  ? f  30.  The  weight  of  lead 
pipe?f  31.  The  weight  of  a  brick  wall  ?f  32.  The  weight  of  a 
wooden  bridge  ?f  33.  The  weight  of  wire  cables  ? f  34.  The 
weight  of  a  ship  in  the  water  ? 

35.  How  many  days  from  June  3d  to  August  19th  ?  36.  From  May 
24th  to  September  9th  ?  37.  From  Jan.  7th,  1848,  to  March  6th"? 
38.  From  Feb.  27th,  1859,  to  Nov.  llth?  39.  Give  the  shortest 
method  that  you  know  for  finding  the  number  of  days  between 
any  given  dates.  40.  If  you  give  a  note  on  Monday,  payable  in 
90  days,  on  what  day  of  the  week  will  it  become  due  ?s  41.  On 
what  day  would  a  note  at  60  days  become  due,  if  given  on  Satur 
day?  42.  On  Thursday?  43.  On  Tuesday?  44.  On  Friday? 
45.  On  Wednesday  ?  46.  A  note  at  30  days,  given  on  Wednesday  ? 
47.  On  Tuesday  ?h  48.  On  what  day  of  the  month  would  a  note 
fall  due,  if  dated  June  27th,  at  30  days  ?  49.  March  9th,  at  60 
days?1  50.  Aug.  1st,  at  90  days?  51.  May  18th,  at  3  months? 


a  Excavations  are  estimated  either  by  the  cubic  yard,  or  by  the 
"square", of  216  cubic  feet. 

b  In  excavating  uneven  ground,  the  depth  may  be  measured  at  seve 
ral  different  points,  and  the  average  of  all  the  measurements  taken. 

0  Gravel  is  estimated  by  the  "  square"  (see  note  a  .) 

d  Stone  is  measured  by  the  "  perch"  of  24 f  cubic  feet. 

e  A  roll  of  paper-hangings  contains  4^  square  yards. 

f  See  Table  of  Specific  Gravities. 

s  93  days  =  13  weeks  and  2  days.  The  note  will  therefore  become 
due  on  Wednesday. 

h  If  the  last  day  of  grace  falls  on  Sunday,  or  on  a  holiday,  the  note 
must  be  paid  on  the  day  previous. 

1  9   and   63   are   72.     Deducting   31   days  for  March,  41  remains. 
Deduct  30  days  for  April,  11  remains.     The  note  will  be  due  May  11. 


§4.]      .  PRACTICE    OF   ARITHMETIC.  27 

52.  October  31st,  at  4  months?1  53.  Sept.  25th,  at  6  months? 
54.  If  the  1st  of  June  falls  on  Monday,  how  many  days'  interest 
shall  I  lose  by  giving  my  note  at  60  days,  dated  May  3d  ?  55.  How 
would  you  determine  the  distance  of  a  cannon,  or  a  thunder 
cloud,  by  observing  the  flash  and  report?15  56.  Suppose  the 
nearest  fixed  star  to  be  suddenly  destroyed,  how  long  would  its 
light  continue  to  reach  us  ?c  57.  How  long  would  it  take  a  loco 
motive,  at  the  rate  of  30  miles  an  hour,  to  travel  the  same  dis 
tance  that  light  goes  in  a  single  second  ?  58.  To  travel  as  far 
as  the  distance  from  the  earth  to  the  sun  ?d  59.  How  far  does 
the  earth  move  in  its  orbit,  while  a  ray  of  light  is  corning  from 
the  sun  ?e 

60.  Can  you  tell  why  a  difference  of  15°  in  longitude,  makes  an 
hour's  difference  in  time?f  61.  In  latitudes  where  a  degree  of 
longitude  is  equivalent  to  45  miles,  how  far  must  you  travel  to  find 
10  seconds  difference  of  time  ?  62.  If  you  travel  east,  will  3*011 
find  the  time  earlier  or  later  than  in  the  place  from  which  you 
start  ?  63.  Why  ?  64.  When  it  is  noon  at  Philadelphia,  what 
time  is  it  at  New  York  ?  65.  Boston  ?  66.  St.  Louis  ?  67.  Lon 
don?  68.  Paris?  69.  Washington ?  70.  Portland?  71.  St.  Peters 
burg  ?  72.  Canton  ?  73.  Astoria  ?  74.  Suppose  the  chronometer 
of  a  vessel  to  be  regulated  by  the  Boston  time,  in  what  longitude 
is  the  vessel,  if  the  sun  passes  the  meridian  at  half-past  nine  by 
the  chronometer  ?  75.  If  an  unbroken  telegraph  wire  should  be 
extended  from  Boston  to  Oregon  City,  at  what  time  could  the  peo 
ple  at  the  latter  place  hear  of  a  transaction  that  occurred  in 
Boston  at  20  minutes  past  3  P.  M.  ?s  76.  What  place  would  the 
same  news  reach  by  telegraph,  at  noon  of  the  same  day  ?  77.  If 
your  watch  keeps  accurate  time,  and  is  correct  when  you  leave 


a  A  note  for  months,  falling  due  on  the  31st,  is  considered  as  due 
on  the  last  day  of  the  month. 

b  Sound  moves  at  the  rate  of  a  mile  in  about  5  seconds. 

c  The  distance  of  the  nearest  fixed  star  is  upwards  of  21000000000000 
miles.  Light  moves  at  the  rate  of  192000  miles  in  a  second. 

d  The  mean  distance  of  the  earth  from  the  sun,  is  about  95  million 
miles. 

e  The  circumference  of  the  earth's  orbit  is  about  600  million  miles. 

f  How  many  degrees  does  any  point  on  the  earth's  surface  pass  over 
in  a  day  ? 

g  No  perceptible  time  will  elapse  during  the  passage  of  electricity 
over  the  wire. 


28  TEST   QUESTIONS   ON   THE  [ART.  II. 

Boston,  how  would  you  set  a  watch  right  that  is  five  minutes 
slower  than  your  own  when  you  reach  Buffalo  ? 

78.  HOAV  would  you  determine  the  area  of  any  piece  of  land,  if 
you  had  no  measure  but  a  yard-stick  ?  79.  Knowing  the  width  of 
a  rectangular  field,  how  would  you  find  the  length  of  a  strip  that 
would  contain  an  acre  ?  80.  How  would  you  estimate  the  area  of 
a  field  by  pacing?1  81.  If  a  road  3  rods  wide,  runs  380  feet 
through  my  land,  what  amount  of  damages  can  I  claim,  the  land 
being  worth  $150  an  acre  ?b  82.  The  length  of  a  certain  field  is 
four  times  the  breadth,  and  the  area  is  ten  acres  ;c  how  many  rods 
of  wall  will  enclose  it  ?  83.  What  must  be  given,  in  order  to 
determine  the  altitude  of  a  triangle  ?  84.  The  base  ?  85.  The 
area? 

86.  How  would  you  estimate  the  contents  of  a  load  or  a  pile  of 
wood  ?d  87.  Of  a  pile  of  boards  ?e  88.  Of  a  pile  of  bricks  ? f 
89.  Of  a  stack  of  hay  ?«  90.  Of  a  heap  of  earth  ?  91.  Of  a  wall  ? 
92.  Name  some  object  that  you  think  would  measure  about  1  foot ; — 

1  yard  ; 3  yards  ; — 1  rod  ; — some  distance  that  you  think  about  a 

furlong.  93.  Draw  a  line  about  1  inch  long ;  3  inches  ;  6  inches; 
9  inches  ;  1  foot ;  4  inches  ;  8  inches.  94.  How  much  do  you  sup 
pose  a  hogshead11  of  water  would  weigh  ?'  95.  If  you  should  wish 
to  weigh  15  pounds,  but  had  mislaid  your  weights,  how  could  you 
form  an  estimate  with  water  ?  96.  Give  an  estimate  of  all  the  di 
mensions  of  the  room  you  are  in  ? 

97.  Suppose  a  note  to  be  given  for  $1000,  interest  payable  semi- 
annually,  to  what  would  it  amount  in  4  years,  at  compound  inter 
est  ?  98.  At  simple  interest  ?  99.  Allowing  simple  interest  on  the 

a  A  pace  is  estimated  at  3  feet.    An  ordinary  step  is  about  2*  feet. 

b  $64.77. 

c  The  field  can  be  divided  into  four  squares,  and  the  side  of  one  of 
those  squares  will  be  the  shorter  side  of  the  field. 

*  Give  the  answer  for  piles  of  regular  and  irregular  shapes. 

e  A  cubic  foot=  12  feet  Board  Measure. 

f  27  bricks  measure  1  cubic  foot.  The  number  of  bricks  required 
to  build  a  house,  may  be  estimated  by  dividing  the  number  of  cubic 
feet  in  the  walls  by  .04. 

s  The  area  of  a  circle  may  be  conveniently  found  by  squaring  .8  of 
the  diameter.  Hay  is  sometimes  sold  in  the  stack,  by  the  cubic  foot. 
400  feet  of  trodden  hay,  weigh  about  one  ton. 

h  A  hogshead  of  measure,  is  intended. 

1  A  cubic  foot  of  water  weighs  about  1000  ounces. 


§4.]  PRACTICE    OF   ARITHMETIC.  29 

principal,  and  also  on  the  interest  after  it  becomes  due  ?a  100. 
Allowing  compound  interest  when  notice  is  given,  supposing  the 
debtor  to  be  notified  at  the  end  of  each  year  ?a  101.  How  would 
you  determine  the  face  of  a  note,  that  would  yield  $1000  in  3 
years,  at  simple  interest  ?b  102.  At  compound  interest  ?b  103. 
At  annual  interest,  allowing  simple  interest  on  the  interest  ?b 
104.  How  would  you  determine  the  face  of  a  note  to  be  discount 
ed  at  bank,  in  order  to  obtain  any  required  sum  ?b 

105.  How  many  shillings  are  equivalent  to  .1  of  a  pound  ?  106. 
To  what  decimal  of  a  pound,  is  1  shilling  equivalent  ?  107.  To 
what  decimal  of  a  pound,  are  24  farthings  equivalent  ?  108.  Then 
how  many  must  you  add  to  any  given  number  of  farthings,  to  re 
present  their  value  in  thousandths  of  a  pound  ?  109.  Can  you  give 
any  rule,  deduced  from  the  answers  to  the  preceding  questions, 
for  reducing  shillings,  pence  and  farthings,  to  the  decimal  of  a 
pound,  by  inspection  ?  110.  Can  you  reverse  the  process,  and  give 
a  rule  for  reducing  any  decimal  of  a  pound  to  shillings,  pence  and 
farthings,  by  inspection  ? 

111.  In  computing  interest  at  6  per  cent.,  in  how  many  months 
will  an  investment  gain  .01  of  itself  ?  112.  In  how  many  days  will  it 
gain  .001  ?  113.  Can  you  give  a  convenient  rule  for  determining  by 
inspection,  the  interest  on  $1  at  5  per  cent.,  6  per  cent.,  and  7  per 
cent.,  for  any  given  time  ?  114.  Give  similar  rules  for  finding  the 
amount,  and  the  present  worth  of  $1  for  any  given  time.  115. 
Can  you  think  of  any  other  abbreviations  in  computing  interest  ?c 


a  The  laws  of  different  states  vary  with  regard  to  compound  interest. 
In  many  places  it  is  collected  on  all  notes;  in  some  cases  the  note  is 
renewed  each  year  and  the  interest  is  included  in  the  new  note  ;  by 
the  laws  of  some  states,  compound  interest  may  be  collected,  provided 
notice  is  given  when  the  interest  becomes  due  ;  and  in  some  states, 
simple  interest  is  allowed  on  the  principal,  and  also  on  all  the  interest 
from  the  time  each  payment  becomes  due  till  the  final  settlement. 

b  After  finding  how  much  $1  would  yield  in  the  given  time,  how 
would  you  find  the  number  of  dollars  required  to  yield  the  desired 
amount  ? 

c  The  following  rule  will  be  found  very  convenient  in  computing 
interest  on  notes  and  accounts,  when  the  rate  is  6  per  cent.  Multiply  } 
per  cent,  of  the  principal  by  one -half  the  even  number  of  months,  and  if 
there  is  an  odd  month,  add  30  to  the  number  of  days.  Divide  the  dai/s  by  6, 
and  multiply  .001  of  the  principal  by  the  quotient.  If  there  are  any  re- 


30  TEST   QUESTIONS    ON    THE  [ART.  II. 

116.  If  you  intrust  a  certain  sum  to  a  factor,  to  cover  the  whole 
amount  of  his  purchases  and  commission,  how  would  you  find  the 
amount  he  can  lay  out?  117.  In  selling  a  certain  invoice  of 
merchandise  at  wholesale,  a  discount  of  15  per  cent,  was  made 
from  the  retail  price.  The  clerk,  in  making  out  the  account,  cal 
culated  15  per  cent,  on  the  whole  cost,  and  deducted  it  from  the 
bill.  Could  you  have  told  him  any  readier  mode  of  obtaining  the 
result  ? 

118.  How  would  you  determine  the  present  value  of  a  widow's 
dower  ?a  119.  The  value  of  a  pension,  payable  during  the  life  of 
one  or  more  individuals?*  120.  The  amount  that  should  be 
annually  contributed  to  secure  the  payment  of  any  desired  sum, 
at  a  person's  decease  ?a  121.  The  amount  of  an  annual  payment, 
for  securing  a  weekly  contribution  during  illness  ?b  122.  The 
amount  of  a  legacy,  sufficient  to  erect  a  bridge  and  provide  funds 
for  all  the  repairs  that  it  will  ever  probably  need  ?  123.  The 
amount  of  weekly  savings  necessary,  to  make  a  young  man  worth 
$5000  in  20  years  ?  124.  The  amount  of  weekly  savings  necessary 
for  cancelling  a  debt,  with  all  the  interest,  in  any  given  time  ? 

125.  How  would  you  reduce  Sterling  to  Federal  Money,  at  9 
per  cent,  premium?  126.  Federal  to  Sterling  Money,  at  7?,  per 
cent,  premium  ?  127.  How  would  you  find  the  value  of  stock  at  a 
discount  of  27  per  cent.  ?  128.  At  an  advance  of  18f  per  cent.  ? 
129.  How  are  interest  and  discount  computed  in  Banks?  130. 
How  do  you  compute  percentage  on  English  Money  ? 

131.  If  you  have  the  diameter  of  a  circle  given,  how  would  you 
find  the  diameter  of  a  circle  that  is  16  times  as  large  ?  132.  J  as 
large?  133.  2 .56  times  as  large  ?  134.  49  times  as  large  ?  135.  If 
you  know  the  area  of  a  field,  what  would  be  the  area  of  a  similar 
field,  each  side  of  which  is  £  as  long  ?  136.  3  times  as  long  ?  137.  7 
as  long?  138.  If  a  ball  2  inches  in  diameter,  weighs  1£  pounds, 
what  would  be  the  weight  of  a  similar  ball,  6  inches  in  diameter  ? 
139.  What  would  be  the  diameter  of  a  similar  ball  that  would 
weigh  96  pounds  ?  140.  If  a  tree  1  foot  in  diameter,  yields  2 
cords  of  wood,  how  much  wood  is  there  in  a  similar  tree  that  is  3 


mriininrr  dni/z,  take  as  many  60(hs  of  1  per  cent.  Add  the  products  thus 
obtained,  and  Uteir  sum  will  be  the  interest  at  &  per  cent. 

*•  All  these  questions  are  solved  by  estimating  the  probable  duration 
of  life.  After  that  is  determined,  what  remains  to  be  done  ? 

b  The  average  amount  of  sickness  is  supposed  to  be  known. 


§4.]  PRACTICE    OF    ARITHMETIC.  31 

feet  6  inches  in  diameter  ?  141.  If  a  hollow  sphere  3  feet  in  dia 
meter  and  2J  inches  thick,  weighs  1 2  tons,  what  are  the  dimen 
sions  of  a  similar  sphere  that  weighs  324  tons  ? 

142.  Knowing  the  original  cost  of  any  article,  how  would  you 
determine  the  price  at  which  it  must  be  sold,  in  order  to  gain  any 
given  per  cent.  ?  143.  How  do  you  find  the  percentage  gained  or 
lost  in  any  transaction  ?  144.  The  dividend  that  a  bankrupt  can 
pay  upon  each  dollar  of  his  debts  ?  145.  The  percentage  of  in 
crease  in  the  population  of  a  place?  146.  The  original  cost,  by 
knowing  how  much  per  cent,  is  gained  or  lost  by  selling  at  a  given 
rate  ?  147.  The  entire  value,  by  knowing  the  value  of  any  given 
percentage  ? 

148.  How  would  you  determine  the  time  at  which  a  debt  could  be 
cancelled  by  a  note  for  any  particular  amount?  149.  The  time  at 
which  several  debts  can  be  cancelled  by  a  single  payment?  150 
The  amount  of  interest  due  on  an  unsettled  account,  there  being 
debits  and  credits  embraced  in  the  account?  151.  The  average 
time  for  settling  an  account,  in  which  there  are  charges  with 
different  times  of  credit?  152.  Can  you  give  more  than  one 
method  for  averaging,  or  equating  an  account  ? 

153.  Why  do  we  begin  at  the  left  hand  in  division?  154.  Do 
you  know  of  more  than  one  mode  of  proof  for  each  of  the  simple 
rules.  155.  What  is  meant  by  the  Arithmetical  Complement  of  a 
number?  156.  Can  you  perform  a  number  of  additions  and  sub 
tractions  at  a  single  operation,  by  using  the  Arithmetical  Comple 
ment?  157.  Can  you  multiply  by  three  or  more  figures  at  a 
single  operation? 

158.  How  do  you  find  the  cost  of  articles  sold  by  the  hundred 
or  thousand  ?  159.  Knowing  the  difference  of  longitude  between 
two  places,  how  do  you  find  their  difference  of  time  ?  160.  Know 
ing  the  difference  of  time,  how  do  you  find  the  difference  of  longi 
tude  ?  161.  If  d  represents  the  diameter  of  a  circle,  c  the  circum 
ference,  and  a  the  area,  how  would  you  find  d,  c  and  a  being 
known?  162.  Knowing  c  and  d,  how  would  you  find  a?  163° 
Knowing  a  and  d,  how  would  you  find  c?  164.  How  would  you 
find  the  rate  of  insurance,  knowing  the  premium  and  the  amount 
insured  ?  165.  The  amount  necessary  to  insure,  in  order  to  cover 
ths  premium  and  expenses  of  collecting,  in  addition  to  the  loss? 
106.  The  cost  and  rate  per  cent,  of  profit  or  loss  being  given,  how 
would  you  find  the  amount  of  profit  or  loss  ?  167.  The  cost  given, 
how  would  you  find  the  selling  price  to  gain  or  lose  a  specified 


82  TEST   QUESTIONS.  [ART.  II. 

rate  per  cent.  ?  108.  Cost  and  selling  price  given,  how  would  you 
find  the  rate  per  cent,  of  profit  or  loss?  169.  Selling  price  and 
rate  of  profit  or  loss  given,  how  would  you  find  the  cost?  170. 
From  the  prime  cost  how  would  you  find  the  selling  price  so  as  to 
gain  any  proposed  percentage,  and  allow  a  discount  for  ready 
money  ? 

171.  How  would  you  estimate  the  quantity  of  grain  in  a  rectan 
gular  bin?  172.  In  a  circular  bin?  173.  In  a  pile  against  the 
side  of  a  building?  174.  In  a  pile  in  the  corner  of  a  building? 
175.  How  would  you  find  the  true  weight,  by  a  pair  of  false 
scales  ?  176.  How  would  you  find  the  value  of  any  estate,  know 
ing  its  annual  rent?  177.  Knowing  the  cost,  rent,  and  annual 
outlay  for  taxes  and  repairs,  how  would  you  find  the  rate  of 
interest  that  any  estate  yields?  178.  How  would  you  find  what 
quantity  of  stock  may  be  purchased  for  any  given  sum,  allowing 
for  brokerage?  179.  How  would  you  find  the  rate  of  interest 
gained  by  money  invested  in  stocks  at  any  given  price  ?  180.  How 
would  you  find  what  sum  must  be  laid  out  in  any  kind  of  stock  to 
produce  a  given  annual  income  ? 

181.  What  must  be  given,  and  in  what  manner  would  you  pro 
ceed,  to  determine  the  area  of  a  rectangular  field  ?  182.  To  deter 
mine  either  side?  183.  To  find  either  side  of  a  right  angled 
triangle?  184.  To  find  the  area  of  the  surface  of  a  sphere? 
185.  The  solidity  of  a  sphere  ?  186.  The  solidity  of  a  cone  ?  187. 
Of  a  cylinder?  188.  How  would  you  find  the  area  of  an  irregu 
lar  field?  189.  The  solidity  of  an  irregular  body?  190.  The 
specific  gravity  of  a  body?  191.  The  tonnage  of  a  ship?  192. 
The  contents  of  a  pan  with  slanting  sides  ?  193.  Of  a  cylindrical 
pail?  194.  Give  the  dimensions  of  a  box  that  would  hold  a 
bushel?  195.  A  peck?  196.  A  wine  gallon  ?  197.  A  wine  quart? 
198.  A  beer  gallon  ?  199.  A  gill  ?  200.  For  what  purpose  is  the 
hogshead  measure  used?  201.  How  would  you  find  the  area  of 
a  roof  that  would  fill  a  cistern  of  given  dimensions,  with  a  fall  of 
£  inch  of  rain  ?  202.  What  is  meant  by  gross  weight  ?  203.  Net 
weight  ?  204.  Tare  ?  205.  What  is  meant  by  a  common  year  ? 
206.  A  sidereal  or  periodic  year  ?  207.  A  leap  year  ?  208.  A  civil 
year  ?  209.  How  would  you  find  the  number  of  gallons  equivalent 
to  one  foot  in  depth  of  a  cistern?  210.  The  number  of  plants  in  an 
acre  of  ground,  their  distance  apart  being  given  ?  211.  What  do 
you  understand  by  an  engine  of  forty  horse  power  ? 


§5.] 


MISCELLANEOUS   EXAMPLES. 


33 


III.    THE  FUNDAMENTAL  RULES. 

N.  B.  The  pupil  should  solve  all  the  questions  that  lie  can, 
mentally.  Let  him  never  use  the  slate  in  obtaining  an  answer, 
except  when  it  is  absolutely  necessary. 

5.  EXAMPLES  IN  ADDITION  AND  SUBTRACTION. 

1-4.  Find  from  the  following  table,  the  population  and 
extent  of  the  globe,  according  to  each  of  the  authorities 
mentioned. a 


^        ,  _.  .  .           |  According  to  Balbi. 

Weimar  Aim.  1840. 

of  the  Globe. 

Population. 

English 

Sq.  Miles. 

Population. 

English 
Sq.  Miles. 

Europe      -    -    - 

227700000 

3700000 

233240043 

3807195 

Asia     -    -    -    - 

390000000 

16045000 

608516019 

17805146 

Africa  -     -     -     - 

GOOOOOOO 

11254000 

101498411 

11647428 

America   -     -     - 

39000000 

14730000 

48007150 

13542400 

Oceanica  -    -    - 

20300000 

4105000 

1834194 

3347840 

5.  The  population  of  China  in  1743,  according  to  the 
French  missionaries,  was  150029855 ;  in  1825,  according 
to  Dr.  Morrison,  352866002.  What  is  the  difference 
between  these  two  estimates  ? 

6-9.  The  skull  has  8  bones,b  the  face  14,c  the  ear  4,d 
the  tongue  l,e  and  there  are  32  teeth/  How  many  bones 


.   a  American  Almanac,  1842. 

b  Osfrontis,  two  ossa  parietalia,  two  ossa  temporum,  os  occipitis,  os 
sphenoides,  and  os  ethnoides. 

c  Two  ossa  maxillaria  superiora,  two  ossa  nasi,  two  ossa  unguis, 
two  ossa  malarum,  two  ossa  palati,  two  ossa  spongiosa  inferiora,  the 
vomer,  and  the  os  maxillare  inferius. 

d  Malleus,  incus,  os  orbiculare,  and  stapes. 

e   Os  hyoides. 

(  Sixteen  in  each  jaw,  viz  :  four  incisores,  two  cuspidati,  four  bicus- 
pides,  and  six  molares. 


34  FUNDAMENTAL   RULES.  [ART.  HI. 

in  the  whole  head  ?  Which  are  the  most  numerous  ?  By 
how  many  do  they  exceed  all  the  others  ?  If  you  subtract 
each  class  of  the  bones,  named  above,  from  the  entire  num 
ber,  how  many  will  be  left  after  each  subtraction  ? 

10-13.  In  the  trunk  there  are  24  spinal  bones,a  24  ribs, 
the  sternum,  or  breast-bone,  the  os  sacrum,  the  os  cocci/gis, 
and  two  ossa  innominata.  Required  the  number  in  the 
entire  trunk  ?  In  the  trunk  and  head  together  ?  How 
many  more  in  the  trunk  than  in  the  skull  ?  How  many 
more  in  the  head  than  in  the  trunk  ? 

14-17.  Each  of  the  upper  extremities  contains  the  follow 
ing  bones,  viz :  2  in  the  shoulders,13  3  in  the  arm,c  8  in 
the  carpus,  or  wrist,  5  in  the  metacarpus,  or  palm,  2  in  the 
thumb,  and  3  in  each  of  the  fingers.d  How  many  in  all  ? 
How  many  in  both  of  the  upper  extremities  ?  In  the  head, 
trunk,  and  upper  extremities  ?  How  many  more  in  one  of 
the  upper  extremities  than  in  the  face  ? 

18-21.  Each  of  the  inferior  extremities  contains  the  fol 
lowing  bones,  viz :  4  in  the  leg,6  7  in  the  tarsus,  or  ankle, 
5  in  the  metatarsus,  or  foot,  and  3  in  each  of  the  toes, 
except  the  great  toe,  which  has  but  two.  Required  the 
number  in  each  of  the  lower  extremities  ?  In  both  ?  In 
all  the  extremities  ?  In  the  whole  body  ? 

Ans.  to  the  last,  240. f 

22-25.  How  many  bones  in  the  whole  body,  besides 
those  of  the  head  ?  Besides  those  of  the  trunk  ?  Of  the 
upper  extremities  ?  Of  the  lower  extremities  ? 

26-29.  The  skin  of  the  cranium  has  3  principal  mus- 

The  24  true  vertebrae,  7  cervical,  12  dorsal,  and  5  lumbar. 

The  clavicle  and  scapula. 

Humerus,  radius,  and  ulna. 

Called  phalanges. 

Os  femoris,  tibia,  fibula,  and  patella. 

Besides  the  bones  enumerated,  are  the  sesamoid  bones,  which 
vary  in  number.  Marjolin  counts  five  for  each  of  the  upper,  and  three 
for  each  of  the  lower  extremities,  making  two  hundred  and  fifty-six 
in  the  whole  body. —  Wistar's  Anatomy. 


§5.]  MISCELLANEOUS   EXAMPLES.  35 

cles,a  each  ear  has  4,  the  lids  of  each  eye  2,  each  eye  6? 
the  nose  2,  the  mouth  19,  the  tongue  8,  and  the  lower  jaw 
has  4  pairs.  How  many  in  all  ?  How  many  less  than  the 
whole  does  the  mouth  contain  ?  How  many  less  than  half  ? 
What  is  the  difference  between  the  number  of  bones  and 
the  number  of  muscles  in  the  head  ? 

30-33.  The  neck  and  throat  contain  50  principal  mus 
cles,  the  trunk  116,  each  of  the  superior  extremities  46, 
and  each  of  the  inferior  extremities  51.  How  many  in  all  ? 
How  many  in  the  entire  body  ?  How  many  less  in  the 
head  than  in  the  rest  of  the  body  ?  How  many  more  muscles 
than  bones  in  the  whole  system? 

34-37.  Find  the  difference  between  the  following  num 
bers,  commencing  the  subtraction  at  the  left  hand : b 
84.9108757944362  and  190027.08;  67490083574.00882 
and  3375109884726.37095 ;  $16438  and  $9728.87758403 ; 
7  and  .0019547998027184920625. 

38-44.  Determine  from  the  following  table,  the  entire 
population  of  the  United  States  and  Territories,  at  each  of 
the  given  dates.  In  obtaining  the  results,  add  two  columns 
of  figures  at  a  time.0 

a  One  pair  and  one  single  muscle.  Most  of  the  muscles  are  found 
in  pairs,  one  on  each  side  of  the  body.  There  is  sometimes  a  slight 
difference  in  the  number,  in  different  individuals. 

b  In  beginning  to  subtract  at  the  left  hand,  if  at  any  point  the  re 
maining  figures  of  the  subtrahend  are  greater  than  those  of  the  minu 
end,  we  must  add  one  to  the  figure  we  are  subtracting.    For 
example,  in  subtracting  92824  from  164809,  say  9  from  16      164809 
leaves  7  ;  then  as  824  is  greater  than  809,  say  3  from  4  leaves 
1  ;  as  24  is-greater  than  09,  say  9  from  18  leaves  9  ;  2  from  10        71935 
leaves  8,  4  from  9  leaves  5.     Let  the  pupil  explain  the  ra 
tionale  of  this  process. 

c  The  teacher  will  find  this  a  valuable  exercise,  to  be  performed 
aloud,  at  the  recitation  of  the  class.  In  adding  two  or  more  columns 
at  once,  it  will  be  found  most  convenient  to  commence  at  the  left 
hand,  and  observe  at  each  step  whether  there  will  be  one  to  bring 
from  the  right  hand  figures.  Thus  in  adding  972  and  645,  instead  of 
saying  5  and  2  are  7,  4  and  7  are  11,  9  and  6  are  15  and  1  are  16, — say 
9  and  6  and  1  are  16,  7  and  4  are  1 1,  5  and  2  are  7.  After  a  little  prao- 


36 


FUNDAMENTAL  RULES. 


[ART.  III. 


TABLE  OF  POPULATION— FROM  THE  AMERICAN  ALMANAC. 


STATES. 

1790. 

1800. 

1810. 

1820. 

1830. 

1840. 

1850 

Maine 
Vew  Hampshire 
Vermont 
Massachusetts 
Rhode  Island 
Connecticut 
New  York 
New  Jersey 
Pennsylvania 
DelaAvare 
Maryland 
Dist.of  Columbia 
Virginia 
North  Carolina 
South  Carolina 
Georgia 
Alabama 
Mississippi 
Louisiana 
Florida 
Texas 
Kentucky 
Tennessee 
Ohio 
Indiana 
Illinois 
Michigan 
Iowa 
Wisconsin 
Missouri 
Arkansas 
Minesota  Terr. 
Missouri    " 
Oregon       " 
Indian         " 
New  Mexico 
California 

96.540 
41,899 
85.416 
378',717 
69,110 
238,141 
340,120 
184,139 
434,373 
59.098 
319',728 

74«,308 
393,751 
249,073 

82,548 

151,719 
183,762 
154.465 
423^245 
69!  122 
•251,002 
586,756 
211,949 
602^365 
64,273 
341.548 
14;  093 
880,200 
478,103 
345,591 
162,101 

'  8,850 

228,705 
214,360 
217,713 
472,040 
77,031 
262,042 
959.949 
249^555 
810,091 
72.674 
380,516 
24,023 
974,622 
555,500 
415,115 
252,433 
20,845 
40,352 
76,556 

298,335 
244,161 
2a5,764 
523,287 
83,059 
275,202 
1,372,812 
277,575 
1,049,458 
72i749 
407,350 
33,039 
1,065,379 
638.829 
502;~41 
340,987 
127,901 
75,448 
153,407 

399,955 
269,328 
280,652 
610,408 
97,199 
297,665 
1,918,608 
320,823 
1,348,233 
76,748 
447,040 
39,834 
1,211,405 
737,987 
581,185 
516,823 
309,527 
136,621 
215,739 
34,730 

501,793 
284,574 
291,948 
737,699 
108,830 
309,978 
2,428,921 
373J306 
1,724,033 
78,085 
469,232 
43,712 
1,239,797 
753,419 
594,398 
691  ,392 
590,756 
375,651 
352,411 
54,477 

73,077 
35,791 

220,955 
105,602 
45,365 

4,875 

406.511 
261^727 
230,760 
21,520 
12,282 
4,762 

564,317 
422,813 
581,434 
147,178 
55,211 
8,896 

687,917 
681,904 
937,903 
343,031 
157,455 
31,639 

779,828 
829,210 
1,519,467 
685,866 
4  76  \  183 
212,267 
43,112 
30,945 
383,702 
97,574 

20,845 

66.536 
14,273 

140,445 

30,388 

u  i 

45-57.  Find  the  increase  in  the  population  from  1790  to 
1800;  to  1810;  1820;  1830;  1840;  1850;  from  1800  to 
1810;  1810  to  1820;  1820  to  1830;  1830  to  1840;  1840 
to  1850;  1790  to  1820;  1820  to  1850. 

58-181.  Find  the  amount  of  each  of  the  following  col 
umns,  and  also  the  total  amount  of  receipts  and  expendi 
tures  in  each  year.  Add  two  columns  of  figures  at  a  time. 

tice  the  pupil  will  say  at  once,  972  and  645  are  1616,  as  naturally  as  he 
would  say  8  and  6  are  14,  without  stopping  to  count  his  fingers.  The 
numbers  to  be  added  should  not  be  mentioned  in  performing  the  addi 
tion.  Thus  in  adding  8,  5,  9,  3,  7,  9,  6,  4,  say  8,  13,  22,  25,  32,  41, 
47,  51,  instead  essaying  8  and  5  are  13,  and  9  are  22,  and  3  are  25,  &c. 
In  adding  65,  48,  27,  92,  say  65,  113,  140,  232,  and  proceed  in  like 
manner  in  all  cases. 


§5.] 


MISCELLANEOUS  EXAMPLES. 


37 


STATEMENT  OF  THE  RECEIPTS  AND  EXPENDITURES  OF  THE 
UNITED  STATES  FOR  SIXTY  YEARS. 

FROM    THE    AMERICAN    ALMANAC,   AND    PUBLIC    DOCUMENTS. 


Years. 

RECEIPTS. 

EXPENDITURES. 

Customs. 

Internal  and 
direr):  taxes. 

Lands  and 
Miscellan. 

Civil  and 
Miscellan. 

Army. 

Navy. 

1789-91 

$4,399,473 

$1,083,401 

$835,618 

$570 

179-2 

3,443,071 

$208,943 

654,257 

1,223,594 

53 

1793 

4,255,306 

337,706 

472,450 

1,237,620 

1794 

4,801,065 

274,090 

705,598 

2>33,540 

61,409 

1795 

5,588,461 

337,755 

1,367,037 

2,573,059 

410,562 

1796 

6,567,988 

475,290 

$4,836 

772,485 

1,474,661 

274,784 

1797 

7,549,650 

575,491 

83,541 

1,246,904 

1,194,055 

382,632 

1798 

7.106,062 

644,358 

11,963 

1,111,038 

2,130,837 

1,381,348 

1799 

6,610,449 

779,136 

1,039,392 

2J582.693 

2,858.082 

1800 

9,080,933 

1,543,620 

444 

1,337.613 

2,625,041 

3,448,716 

1801 

10,750,779 

1,582,377 

167,726 

1,114J768 

1,755,477 

2,111,424 

1802 

12,433,236 

828,464 

188,628 

1,462,929 

1,358,589 

'915.562 

1803 

10,479,418 

287,059 

165,676 

1,842.636 

944,958 

1,215;231 

1804 

11,098,565 

101,139 

487,527 

2,19i;009 

1,072,017 

1,189,833 

1805 

12,936,487 

43,631 

540,194 

3,768,588 

991,136 

1,597.500 

1806 

14,667,698 

75,865 

765,246 

2,891,037 

1,540,431 

1,649;641 

1807 

15,845,522 

47,784 

466,163 

1,697,897 

1,564,611 

1,722,064 

1808 

16,363,550 

27,370 

647,939 

1,423,286 

3,196,985 

1,884,068 

1809 

7,296,021 

11,562 

442,252 

1,215,804 

3,771,109 

2,427,759 

1810 

8,583,309 

19,879 

696,549 

1,101,145 

2,555,693 

1,654,244 

1811 

13,313,223 

9,962 

1,040,238 

1,367,291 

2,259,747 

1,965,566 

1812 

8,958,778 

5,762 

710,428 

1,683,088 

12,187,046 

3,959,365 

1813 

13,221,623 

8,561 

835,655 

1,729,435 

19,906,362 

6,446,600 

1814 

5,998,772 

3,882,482 

1,135,971 

2,208,029 

20,608,366  :  7.311,291 

1815 

7,282,942 

6,840,733 

1,287,959 

2,898,871 

15,394,700 

8,660,000 

1816 

36,306,875 

9,378,344 

1,717,985 

2.989,742 

16,475,412 

3,908.278 

1817 

26,283,348 

4,512,288 

1,991,226 

3^518,937 

8,621,075 

3,314;598 

1818 

17,176:335 

1,219,613 

2,606,565 

3,835,839 

7,019,140 

2,953,695 

1819 

20,283,609 

313.244 

3,274,423 

3,067,212 

9,385,421 

3,847,640 

1820 

15,005,612 

137,847 

1,635,872 

2,592.022 

6,154,518     4,387,990 

1821 

13,004,447 

98,377 

1,212,966 

2,223;  122 

5,181.114     3,319,243 

1822 

17,589,762 

88.617 

1,803,582 

1,967,996 

5,635;  187 

2,224,459 

1823 

19,088,433 

44,580 

916,523 

2,022,094 

5,258,295 

2,503,766 

1824 

17,878,326 

40,865 

984,418 

7,155,308 

5,270,255 

2,904,582 

1825 

20,098,714 

28,102 

1,216,090 

2,748,544 

5,692,831 

3,049,084 

1826 

23,341,332 

28,228 

1,393,785 

2,600,178 

6,243,236 

4,218,902 

1827 

19,712,283 

22,513 

1,495,945 

2,314,777 

5,675,742 

4,263,878 

1828 

23,205,524 

19,671 

1,018,309 

2,886,052 

5,701,203 

3,918,786 

1829 

22,681,966 

25,838 

1,517,175 

3,092,214 

6,250,530 

3,308,745 

1830 

21,922,391 

29,141 

2,329,356 

3,228,416 

6,752,689 

3,239,429 

1831 

24,224,442 

17,440 

3,210,815 

3,064,346 

6,943,239 

3,856,183 

1832 

28,465,237 

18,422 

2,623,381 

4,574,841 

7,982,877 

3,956,370 

1833 

29,032,509 

3,153 

3,967,682 

5,051,789 

13,096,152 

3,901,357 

1834 

16,214,957 

4,216 

4,857,601 

4,399,779 

10,064,428 

3,956,260 

1835 

19,391,311 

14,723 

4,757,601 

3,720,167 

9,420,313 

3,864,939 

1836 

23,409,940 

1,099 

4,877,180 

5,388,371 

18,466,110 

5,800,763 

1837 

11,169,290 

6,863,556 

5,524,253 

19,417,274 

6,852,060 

1838 

16,158,800 

3,214,184 

5,666,703 

19,936,312 

5,975.771 

1839 

23,137,925 

7,261,118 

4,994,562 

14,268,981 

6,223^03 

1840 

13,499,502 

3,494,356 

5,581,878 

11,621,438 

6,124,456 

1841 

14,487,217 

1,470,295 

6,490,881 

13,704,882 

6,001,077 

1842 

18,187,969 

1,456,058 

6,775,625 

9,188,469 

8,397,243 

1843  a 

7,046,844 

1,018,482 

2,867,289 

4,158,384 

3,672,718 

1844  b 

26,183,571 

2,320,948 

5,231,747 

8,231,317 

6,496,991 

1845  b 

27,528,113 

2,241,021 

5,618,207 

9,533,203 

6,228,639 

1846  b 

26,712,668 

2,786,579 

6,783,000 

13,579,428 

6,450,862 

1847  b 

23,747.864 

2,598,926 

6,715,854 

41,281,606 

7,931,633 

1848  b 
1849  b 

31,757,071 

3,679,680 

5,585,070 

27,820,163 

9,406,737 

•  6  months  of  1843. 


b  For  the  year  ending  June  30. 


88 


FUNDAMENTAL  RULES. 


[ART.  HI. 


©.  EXAMPLES  IN  ADDITION,  SUBTRACTION,  MULTIPLI 
CATION,  AND  DIVISION. 

1.  Invoice  of  6  bales  of  Dry  Goods : — 

Boston,  November  27,  1849. 
Messrs.  Thompson  &  Allen, 

Bought  of  William  Mansfield,* 
T.  &  A.    A  bale  containing 

15  pieces  Lint  Strelitz  Osnaburgs, 
each  130  yds.,     @  lOc. 
"Wrapper,  20  yds.,  @  6c. 
Bale,  cording  and  packing, 


No.  35. 


36  to  40 


Five  bales,  containing 

No.  36,  464  yds. 
No.  37,  481 
No.  38,  437 
No.  39,  475 
No.  40,  470 


2327    "  @  He. 
Bales,  cording  and  packing, 


1 

5 

25 

00 

$458 

42 


2.  Bill  and  Receipt : — 

Philadelphia,  March  25,  1848. 
Benjamin  Stabler, 

To  John  Farmer,  Dr. 

To      4  tons  of  hay,        at  $25  50 
115  bushels  of  oats,  "  30 

95      «  corn,  "  94 


Received  payment, 


JOHN  FARMER. 


*  The  items  should  be  extended  in  the  left  hand  column,  and  the 
amounts  of  each  package  (or  all  the  packages  that  are  included  to 
gether,  as  in  Nos.  36  to  40),  carried  out  in  the  right  hand  column. 
This  is  the  usual  form  of  an  American  invoice.  The  pupil  should  give 
the  amount  for  each  of  the  items  that  is  left  blank. 


§6.]  MISCELLANEOUS   EXAMPLES. 

3.  Bill,  unreceipted  : — 
George  Lenox, 


39 


New  York,  July  7,  1849. 


1  doz.  Long  Shawls, 

3  pieces  Sheeting,  30,  31,  33  yds. 

1  piece  Mousseline  de  Laine,  28  " 

2  pieces  Broadcloth,  35  and  40  " 
6  doz.JLinen  Hdkfs. 

12  doz.  Cotton  Half  Hose, 


Bought  of  Carlisle  £  Williams, 


@ 


$6.75 

.09 

.17 

4.75 

.25 

1.50 


4  months. 
4.  Statement  of  Account : — 


$486  47 


Mason,  Hamilton  &  Co. 


Albany,  Jan.  5,  1849. 


1848. 


To  Johnson  &  Brooks,         Drs. 


Aug.  27. 

To  Stoves,  &c.,  per  bill  rendered,  6  mo. 

$843  75 

Sept.  5. 

(.          «                             <( 

u 

375 

69 

"    13. 

(.          a                              u 

u 

11850 

"    30. 

a          a                            (( 

a 

97 

11 

Oct.    8. 

«          «                             a 

t( 

103 

88 

"    23. 

«          ft                            (( 

(( 

491 

35 

Nov.  19. 

"          «                             (( 

t( 

85 

07 

Contra,  Cr. 

Oct.    1. 

By  Cash, 

450.00 

Nov.   7. 

"   acceptance  of  draft, 

200.00 

Dec.  12. 

"   note  at  4  months, 

562.50 

Balance, 

$ 

Due  by  average,  June  21,  E.  ^ 

fc  0.  E.a 

5.  The  population  of  the  Chinese  Empire  has  been  es 
timated"  as  follows:  China  proper,  148897000;  Corea, 
8463000;  Thibet  and  Boutan,  6800000 ;  Mandshuria,  Mon 
golia,  &c.,  9000000 ;  Colonies,  1000000.  Its  territory  con-. 


a  Errors  and  omissions  excepted. 

b  Murray's  Encyclopaedia  of  Geography. 


40  FUNDAMENTAL   RULES.  [ART.  HI, 

tains  about  5850000  square  miles.     Required  the  entire 
population,  and  the  number  of  inhabitants  to  a  square  mile. 

6.  Bill  of  Lading  :— 

Sj)fpj)etJ,  in  good  order  and  condition,  by  Norton  &  Phillips, 

in  and  upon  the  Brig  called  the  Margaret,  whereof 

3fc  1  to  8      Meriam  is  master  for  this  present  voyage,  and  now 

B.  Van  Pelt,    lying  in  the  port  of  Charleston,  and  bound  for  New 

Care  of        York, 

L.  Haines,  Three  Soxes  and  Five  Bales  of  Goods, 

116  Broadway,  Weight  1835  Ibs. 

New  York. 

Being  marked  and  numbered  as  in  the  margin,  to 
be  delivered  in  the  like  good  order  and  condition, 
at  the  aforesaid  port  of  New  York  (the  danger  of 
the  seas  only  excepted),  unto  Benjamin  Van  Pelt, 
or  to  his  assigns,  he  or  they  paying  freight  for  the 
said  goods,  at  the  rate  of  28  cts.  per  100  Ibs.,  with 
Primage  and  Average  accustomed. 

IN  TESTIMONY  WHEREOF,  the  Master  or  Purser 
of  the  said  Brig  hath  affirmed  to  three  Bills  of 
Lading,  all  of  this  tenor  and  date,  one  of  which 
being  accomplished,  the  others  to  stand  void. 

Dated  at  Charleston,  the  30th  day  of  March, 
1849. 

J.  M.  MERIAM. 

Required  the  amount  of  freight  on  the  above  merchan 
dise. 

7.  Receipt  in  full : — * 

Worcester,  June  5th,  1849. 
James  A.  Chase, 

1848.  To  Moses  Allen,  Dr. 

Oct.  27.     To  Mdse.  per  bill  rendered,  27.35 

Nov.  3.       "   Hire  of  Horse  and  Chaise  to  Leicester,    1.50 

1849. 
March  11,    "   1  ton  of  Coal,  9.75 

Received  payment  in  full,  $ 

MOSES  ALLEN. 


*  A  receipt  in  full  is  understood  to  cover  all  demands. 


§6.]  MISCELLANEOUS   EXAMPLES.  41 

8.  Publisher's  Estimate  for  one  thousand  copies  of  a  book  : 
Printing  and  corrections,  $513.00 

Paper  and  certificate  of  copyright,  326.00 

Binding,  175.00 

Advertising,  200.00 


1  copy  to  be  deposited  in  the  Clerk's  Office, 
10  copies  to  the  author, 
989  copies  for  sale  at  $1.75, 
Deduct  cost 


Balance  to  cover  commissions,  interest  on  ^ 
capital,  and  profit  to  author  and  publisher,  v          $516.75 
when  all  are  sold.  j 

9.  Order  for  Goods  :-— 

Providence,  August  18,  1849. 
Messrs.  Anthony  &  Smith, 

Please  ship  to  us  by  first  packet : 
25  cwt.  Sugar  (Brown  Havana)  about  $5.75  per  cwt. 
13  hhd.  W.  I.  Molasses,  $23. 
27  boxes  Louisiana  Oranges,  $3.80. 
Old  Java  Coffee,  say  1300  Ibs.  at  .09. 
Honey,  about  250  gallons  at  .67. 
18  chests  Black  Tea,  $5.50  per  chest. 
49  bbls.  Genesee   Flour,  $4.75. 
And  oblige,  Yours,  &c., 

L.  KENTON  &  KAY. 

What  would  be  the  amount  of  the  above  order,  if  all  the 
articles  were  sent  at  the  prices  affixed  to  them  ? 

10.  Determine  by  the  Roman  Notation,  how  many  years 
elapsed  from  the  discovery  of  America  by  Columbus,  Anno 
Domini  MCCCCXCII,  to  the  adoption  of  the  United  States 
Constitution,  A.  D.  MDCCLXXXIX.a 

*  The  pupil  will  probably  find  no  difficulty  in  adding  or  subtracting 
Roman  numerals,  except  when  such  numbers  as  IV,  IX,  XL,  XC, 


42  FUNDAMENTAL  RULES.  [ART.  III. 

11.  The  Pilgrims  landed  at  Plymouth  in  MDCXX,  and 
Benjamin  Franklin  was  born  LXXXVI  years  afterwards. 
Find  by  the  Roman  Notation,  in  what  year  he  was  born. 

12.  Multiply  by  the  Roman  method,  and  determine  the 
number  of  weeks  that  elapsed  between  the  landing  of  the 
Pilgrims  and  the  birth  of  Franklin,  allowing  LII  weeks  to 
each  year,  and  adding  XII  to  the  product. 

13.  Settlement  of  Account,  exhibiting  different  forms  of 
receipts  : — 

Account  rendered  Jan.  1,  1848. 

Lowell,  Jan.  1,  1848. 
Thomas  Lawrence 

1846.  To  Henry  Appleton,         Dr. 

July  11.     To  Mdse.  per  bill,  $984.32 

"     29.       "  5  bbls.  Flour,  @     $5.25 

Aug.  31.      "  49  yds.  Broadcloth,   @       4.50 
Oct.   17.       "  56  yds.  Carpeting,     @       1.15 

"  Balance  of  interest  from  Jan.  1, 1847,  16.70 


1847. 
Jan.  13. 
Mar.  4. 

Cr. 
By  Cash, 
"   Mdse. 

Balance, 

100.00 
27.42 

127.42 

$1184.75 

occur.     The  difficulty  maybe  removed  by  changing  them  to  the  forms 
IIII,  VIIII,  XXXX,  LXXXX.   Thus,  ifit  were 
required  to   add    DCCXLIV,    CCXCIX,  and    DCCXXXX1III 
MDLXVIII,  writing  them  as  in  the  margin,  we     CCLXXXX VIIII 
commence  at  the  right  hand,  and  say,  eleven  Is    MDLXVIII 

are  equivalent  to  two  Vs  and  I ;  two  Vs  and  two 

to  be  added  from  the  Is  make  four  Vs  or  two        MMDCXI 
Xs ;  nine  Xs  and  two  to  be  added  make  eleven 
Xs,  or  two  Ls  and  X ;  two  Ls  and  two  to  be  added  make  four  Ls,  or 
two  Cs  ;  four  Cs  and  two  Cs  make  six  Cs,  or  one  D  and  C  ;  two  Ds 
and  one  D  are  three  Ds,  or  M  and  D  ;  M  and  M  are  MM. 

In  multiplication,  the  multiplier  maybe  divided  into  any  convenient 
number  of  parts,  and  after  multiplying  by  each  part,  the  several  partial 
products  should  be  added  together. 


§6.]  MISCELLANEOUS   EXAMPLES.  43 

On  settling  the  account,  August  5,  1849,  Thomas  Law 
rence  presented  the  following  receipts,  viz  : — 

I.  A  Receipt  on  account. 

"  Received  of  Thomas  Lawrence  Seventy-five  Dollars,  on 
account.     Lowell,  Jan.  25,  1848. 
$75.  HENRY  APPLETON." 

II.  A  Receipt  through  a  third  person. 

"  Received  of   Thomas  Lawrence,  per  hands  of  James 
Brown,  One  Hundred  and  Ten  Dollars,  on  account. 
$110.  HENRY  APPLETON, 

Lowell,  March  4,  1848.  per  WILLIAM  ALLEN." 

in.  Receipt  for  Note  on  account. 

"  Received  of  Thomas  Lawrence  his  note  dated  July  5, 
1848,  @  8  months  date  my  favor,    for    Seven  Hundred 
and  Fifty  Dollars,  on  account.     Lowell,  July  7,  1848. 
Note  $750.  HENRY  APPLETON." 

The  receipts  being  all  correct,  there  is  found  to  be  due 
to  Henry  Appleton  $78.46  interest,  in  addition  to  the  un 
settled  balance  of  account.  Required  the  amount  that  has 
been  paid,  the  balance  outstanding,  and  the  amount  paid  at 
settlement,  for  which  H.  A.  gives  the  following 

Receipt  in  full. 

"  Received  of  Thomas  Lawrence,  Tir5- 

Dollars,  in  full  of  all  demands  to  this  date. 
$  HENRY  APPLETON." 

Lowell,  August  5,  1849. 

14.  Allowing  365  days  and  6  hours  to  each  year,  what 
length  of  time  would  be  required  to  count  the  number  of 
tons  of  carbonic  acid  contained  in  the  atmosphere,  counting 
3  every  second,  for  12  hours  a  day  ?a 

Am.    111697y.  124d.  5h.  26m.  40sec. 

15.  A  horse-power  is  generally  estimated  as  sufficient  to 
raise  330001bs.  1  foot  high  in  1  minute ;  and  Desaguliers 

a  See  Example  17. 


44  FUNDAMENTAL  RULES.  [ART.  III. 

estimates  the  power  of  1  horse  as  equivalent  to  that  of  5 
men.  According  to  the  latter  estimate,  how  much  would  1000 
men  be  able  to  raise  to  the  height  of  20  feet  in  5  minutes  ? 

Ans.    16500001bs. 
16.  Account  Sales : — 

ACCOUNT  SALES.  8  Hhds.  Molasses,  per  Mayflower,  from  Trinidad, 
for  account  of  Williams  &  Tasistro. 


gals.  qt. 

CHARGES. 

W  T 

3t29,     113 

Insurance  on  $150  at 

29 

30,     112  3 

1  per  cent.            $1.50 

to 

Of> 

31,     115  3 

Policy,                          .50 

oo 

32,     125 

2.00 

33,     119  1 

Freight  and  storage,         19.50 

34,     117 

Duties,                                65.30 

35,     120  3 

Brokerage,                           3.88 
Commission  &  Guaranty,  16.33 

36,     121  2 

Gro.      945    gals. 
Dft.         12      " 

Net  proceeds,  due 

Oct.  5,  1849.             $219.54 

Net       933    gals. 

Boston,  Aug.  28,  1849,    E.  E. 

@  35  cts.    $• 

WELLINGTON,  CARTER  &  Co. 

17.  It  has  been  estimated*  that  the  atmosphere  contains 
3994592925000000  tons  of  nitrogen,  1233010020000000 
tons  of  oxygen,  5287350000000  tons  of  carbonic  acid,  and 
54459705000000  tons  of  aqueous  vapor.  Bequired  the 
weight  of  the  whole,  in  pounds. 

Ans.  11843664  trillion  pounds,  according  to  the  French 
Notation.1* 

a  Griffiths. 

b  The  old  English  Notation  is  still  employed  in  some  works,  but  the 
French  method,  which  is  simpler  and  more  easily  learned,  is  generally 
adopted,  even  by  the  more  modern  English  writers.  In  the  English 
system,  each  period  consisted  of  six  places.  A  billion  was  therefore 
equivalent  to  a  million  million  ;  a  trillion,  was  a  million  billion,  and 
so  on. 


§6.] 


MISCELLANEOUS   EXAMPLES. 


45 


18.  Invoice  of  2  cases  merchandise,  shipped  at  Havre 
on  board  the  ship  Duchesse  d'  Orleans,  Richardson,  Master, 
"bound  for  New  York,  purchased  for  account  and  risk  of 
Messrs.  Hamilton  &  Co.,  of  Philadelphia,  and  to  them  con 
signed. 


225 


226 


One  Case. 

No.  54,  107  S*    214  doz.  Braces  6.b 
55,      1 "        2    "        "       9.50 

Discount  2  °|0  c 


26.05d 


108  Boxes, 
Packing, 


.60 


25.90 


-F.I  3  67. 65 


One  Case. 

No.  72,  29^  58  doz.  Braces,  9.50 

80,  20  "  40    «         "  10. 

87,  15  "  30   "        "  14. 


Discount  8  °|c 

64  Boxes, 
Packing, 


109.70 


.60 


22.80 


Commission  3°|0 


E.  E.  Paris,  25  November,  1849, 
LEROUX  &  CIE. 


80.70 
F.  2770.85 


19.    The  bulk  of  carbonic  acid  produced  by  a  healthy 
adult  in  24  hours,  is  about  15000  cubic  inches,  and  weighs 

*  Cartons. 

b  In  French  currency  the  Franc  is  considered  as  the  unit,  and  Cen 
times  are  written  as  hundredths.  Calculations  are  therefore  made  as  in 
Federal  Money.  Thus  214  doz.  at  6  Fr.  =  1284  Frs. ;  11  doz.  at  1.25, 
would  be  13.75,  &c. 

c  This  character  (  °|0  )  is  an  abbreviation  for  per  cent. 

d  No  fraction  of  a  franc  is  counted,  less  than  5  centimes. 


46  FUNDAMENTAL  RULES.  [ART.  III. 

about  6  ounces.*  If  this  were  the  average  rate,  how  much 
would  be  produced  daily  by  the  population  of  a  town  con 
taining  5000  inhabitants. 

Am.     43402c.  ft.  1344c.  in.=16cwt.  2qr.  271b. 

7.  TABLE  FOR  ADDITIONAL  EXERCISE. 

In  every  kind  of  business,  correctness  and  readiness  in 
the  use  of  figures  are  of  the  first  importance.  These  can 
only  be  obtained  by  practice,  and  to  give  an  opportunity 
for  such  practice,  a  page  of  figures  is  here  inserted,  which 
may  be  so  divided  by  the  teacher  as  to  make  many  thou 
sands  of  examples.  Thus,  in  addition,  the  pupil  may  add 
a  single  column,  or  any  two,  three  or  more  contiguous 
columns,  or  parts  of  columns,  or  the  entire  page ;  and  he 
may  be  exercised  in  adding  two,  three,  or  more  columns  at 
once.b  In  subtraction  (in  which  it  is  recommended  that  he 
should  always  be  required  to  commence  at  the  left  hand),c  he 
may  take  parts  or  the  whole  of  any  two  adjoining  rows,  or 
he  may  subtract,  (without  using  his  slate,)  part  of  the 
figures  in  any  row,  from  the  remaining  figures.  In  multi 
plication  and  division  the  examples  may  be  varied  in  like 
manner  to  any  desired  extent. 

By  this  arrangement  a  whole  class  may  be  engaged  in  the 
same  operation,  without  having  the  question  written  on  the 
board,  and  the  teacher  can  save  the  time  that  might  otherwise 
be  required  for  setting  down  examples  on  the  slate.  Thus, 
should  any  scholar  require  additional  exercise  in  either  of 
the  fundamental  rules,  he  may  be  told  to  add  columns  19, 
20,  and  21,  or  to  subtract  row  15  from  row  14,  prefixing  a 
7  to  the  upper  row;  or  to  multiply  row  7  by  row  11 ;  or  to 
divide  row  3  by  the  first  five  figures  in  row  18 ;  or  he  may 
be  required  to  form  examples  for  himself. 

»  Griffiths.        b  See  note  c,  page  35.        c  See  note  b,  page  35. 


§7.]  TABLE   FOR   ADDITIONAL  EXERCISE.  47 

1234567890123456789012345678 

2  874051749395215780693426792 

3  463791036734597185046803593 
4712595107678977006569554134 

5  531896057402569708964632915 

6  790456157880045678943156366 
7429170836020162968730769357 

8  630690579380083968036970168 

9  975021890679214607359207429 

10  596810752957410215648196310 

11  662071568615627951621084621 

12  146530951936268016951606742 

13  620986814610840168368516103 

14  605636856960683286096241634 

15  296296079638003672956173305 

16  308410466806 4 17794626047906 

17  967086931679067241609706247 

18  164349028160792174267702648 

19  075359116502691769246902619 

20  231670461607156852079510260 

21  290591705793701360792519561 

22  560735790245600982603791572 

23  597590708697908634843973353 

24  586806964804687904790869064 

25  650976082976806470987808085 

26  978496990679680609768907896 

27  787926970608707873016391757 

28  350169517401583925365495818 

29  463701673043681360320726039 

30  036563263596420547390253640 

31  561941357359640473759163031 

32  035704855140267206486251652 

33  054256073465361503672564063 

34  484051330527403842603153054 

35  630484738540168102816214105 

36  601698356109213084572946136 


48  FUNDAMENTAL  RULES.  [ART.  III. 

8.  FRACTIONS  AND  COMPOUND  NUMBERS. 

1-10.  What  part  of  the  entire  number  of  bones  in  the 
human  body  is  contained  in  the  skull?  [See  §5,  Ex.  6-21, 
and  express  all  the  answers  in  the  lowest  terms.]  What 
part  in  the  face  ? — the  ear  ? — the  tongue  ? — the  whole  head  ? 
— the  spine  ? — the  ribs  ? — the  whole  trunk  ? — the  head  and 
trun*k  ? — the  extremities  ? 

11.  Set  down  the  answers  to  the  questions  in  the  prece 
ding  paragraph,  and  add  all  the  fractions  together. 

12-13.  If  you  were  to  count  three  every  second,  for  ten 
hours  a  day,  how  long  would  it  take  you  to  count  a  mil 
lion  ?  How  long  would  it  take  to  count  a  billion,  at  the 
same  rate  ? 

14.  The  population  of  Washington  in  1840  was  23364. 
If  all  the  inhabitants  were  to  commence  counting  at  the 
rate  mentioned  in  the  last  example,  how  long  would  it  take 
them  to  count  a  trillion  dollars  ? 

15.  The  salary  of  the  President  of  the  United  States  is 
$25000  per  annum.     How  many  years'  salary  would  be 
equivalent  to  a  million  dollars  ? 

16.  Pure  silver  is  worth  at  the  mint  $15.50  a  pound. 
How  many  tons  would  it  take  to  be  worth  a  million  dollars  ? 
Express  the  answer  in  tons  and  fractions  of  a  ton,  and  also 
in  tons,  cwt.,  qrs.,  &c. 

17.  Standard  gold  being  worth  $18.60  per  ounce,  what 
weight  of  gold  would  be  required  to  furnish  a  half-eagle  to 
every  inhabitant  of  the  United  States,  estimating  the  popu 
lation  at  21000000  ?     Express  the  answer  as  in  the  last 
example. 

18.  What  is  the  amount  of  the  President's  salary  per 
day  ? — per  hour  ? — per  minute  ? 

19.  The  average  pulsation  of  the  heart  is  in  childhood 
about  105  beats  in  a  minute,  in  youth  90  beats,  in  middle 


§8.]  FRACTIONS   AND    COMPOUND   NUMBERS.  49 

age  75,  and  in  old  age  60.  Estimating  the  duration  of 
childhood  at  14  years,  youth  at  11  years,  middle  age  at 
25  years,  and  old  age  at  25  years,a  how  many  times  would 
the  heart  beat  in  each  period  ?  How  many  times  in  the 
whole  life  ? 

20.  TABLE  OF  DISCOVERIES  AND  IMPROVEMENTS." 

Astronomical  observations  first  made  in  Babylon  .         .  2234 

Lyre  invented            .....  1004 

Sculpture     ......                  .     '  .     *   .  1900 

Agriculture  by  Triptolemus       .....  1600 

Chariots  of  War 


^ 

Alphabetical  letters  introduced  into  Europe      .         .         .  1500 

The  first  ship  seen  in  Greece,  arrived  at  Rhodes  from  Egypt  1485 

Iron  discovered  in  Greece,  by  the  burning  of  Mount  Ida   .  1406 

Seaman's  Compass  invented  in  China   ...                  .  H20 

Gold  and  Silver  money  first  coined  by  Phidon,  king  of  Argos  894 

Parchment  invented  by  Attains,  king  of  Pergamus        .         .  887 

Weights  and  measures  instituted       .         .         .  $69 

First  astronomical  observation  of  an  eclipse         .         .  721 

Ionic  order  used  in  building      ....  650 

Maps  and  globes  invented  by  Anaximander           .         .         .  600 

Sun-dials  invented    .  cr« 

•         •         •         .  ooo 

Signs  of  the  Zodiac,  invented  by  Anaximander     .         .         .  547 

Corinthian  order  of  architecture        .         .         .  540 

First  public  library  established  at  Athens    .         .         .  505 

Silk  brought  from  Persia  to  Greece  .         .         .  325 
The  art  of  painting  brought  from  Etruria  to  Rome,  by  Quintus 

Pictor       ..........  291 

Solar  quadrants  introduced       ....  290 

Mirrors  in  silver  invented  by  Praxiteles        .         .         .  288 

Silver  money  first  coined  at  Rome     .....  269 

Water-clocks         ......  245 

Hour-glass  invented  in  Alexandria   ....  240 

Burning  mirrors  invented  by  Archimedes     .         .         .  212 

First  fabricating  of  glass           .....  200 

a  Encyclopaedia  Americana,  Rees'  Encyclopaedia. 
b  Mcmoria  Technica,   Rees'   Encyclopaedia,  Beckmann,  Scientifio 
American. 

4 


50  FUNDAMENTAL  RULES.        [ART.  HI. 

B.  C. 

Brass  invented 146 

Paper  invented  in  China 105 

Rhetoric  first  taught  at  Rome 87 

Blister-plasters  invented 60 

Julian  year  regulated  by  Csesar 45 

Apple  trees  brought  from  Syria  and  Africa  into  Italy       .  9 

A.  c. 

Vulgate  edition  of  the  Bible  discovered     .         .         .         .  218 

Porcelain  invented  in  China 274 

Water-mills  invented  by  Belisarius 555 

Sugar  first  mentioned,  by  Paul  Eginetta,  a  physician '  .         .  625 

Quills  used  for  writing 636 

Stone  buildings  first  erected  in  England,  by  Bennet,  a  monk    670 

The  system  of  couriers,  or  posts,  invented  by  Charlemagne  808 

Figures  used  by  the  Arabs,  borrowed  from  the  Indians         .  813 

Lanterns  invented  by  king  Alfred 890 

High  towers  first  erected  on  churches       ....  1000 

Musical  notes  invented  by  Guido,  of  Arezzo          .         .         .  1021 

Heraldry  originated 1100 

Distillation  first  practised 1150 

Glass  windows  first  used  in  England         .         .         .         .1180 

Chimneys  built  in  England 1236 

Leaden  pipes  for  conveying  water,  invented     .         .         .  1252 

Magic  lanterns  invented  by  Roger  Bacon     ....  1290 

Tallow  candles  first  used 1290 

Fulminating  powder  invented  by  Roger  Bacon     .         .         .  1290 

Spectacles  invented  by  Spina 1299 

Wind-mills  invented 1299 

Alum  discovered  in  Syria 1300 

Paper  made  of  linen 1302 

Coals  first  used  in  England 1307 

Saw-mills  at  Augsburg 1322 

Woollen  cloths  first  made  in  England        ....  1331 

Gold  first  coined  in  England 1344 

Painting  in  oil  colors 1410 

Muskets  used  in  England 1421 

Pumps  invented 1425 

Printing  invented          .                   1440 

Glass  first  made  in  England 1457 

Wood-cuts  invented 1460 

Almanacs  first  published  in  Buda 1460 


§8.]           TRACTIONS  AND   COMPOUND   NUMBERS.  51 

Printing  introduced  into  England  by  Caxton        .         .         .  1470 

Watches  invented  at  Nuremberg 1477 

Stages  and  post  horses  established                ....  1483 

Tobacco  discovered  in  St.  Domingo           ....  1496 

Shillings  first  coined  in  England 1505 

Stops  in  literature  introduced 1520 

Spinning-wheel  invented  at  Brunswick         ....  1530 

Variation  of  the  Compass  first  observed            .         .         .  1540 

Pins  first  used  in  England 1543 

Needles  first  made  in  England  by  an  East  Indian     .         .  1545 

Sextant  invented  by  Tycho  Brahe 1550 

Lace  knit  in  Germany               1561 

Coaches  first  used  in  England                1580 

Telescopes  invented  by  Jansen 1590 

Stocking  weaving  invented            ......  1590 

Decimal  arithmetic  invented  at  Bruges     ....  1602 

Microscopes  used  at  Naples          ......  1618 

Circulation  of  the  blood  discovered  by  Harvey          .         .  1019 

Coins  made  with  dies  in  England 1620 

Thermometers  invented  by  Drehel 1620 

Barometer  invented  by  Torricelli,  an  Italian    .         .         .  1626 

Newspapers  first  published            ......  1630 

Regular  posts  established  in  London         ....  1635 

Coflee  brought  to  England 1641 

Steam  engines  invented  by  the  Marquis  of  Worcester       .  1649 

Pendulums  for  clocks  invented 1649 

Air-pumps  invented 1650 

Air-guns  invented  by  Outer          .         .         .         .         .         .  1656 

Spring  pocket  watches  invented  by  Dr.  Hook  .         .         .  1658 

Engines  invented  to  extinguish  fires 1663 

Bayonets  invented  at  Bayonne 1670 

Micrometer  invented 1677 

Telegraphs  invented 1687 

New  style  adopted  in  England 1752 

Spinning  frame  by  Arkwright 1761 

Cotton  first  planted  in  the  United  States      .         .         .         .1769 

Steam  engine  improved  by  Watts      .....  1769 

Georgium  Sidus  discovered  by  Herschel       ....  1781 

Power  looms  invented  by  Cartwright         ....  1783 

Steam  cotton-mills  first  erected 1783 

Steam  grist-mills  first  erected 1785 

Stereotype  printing  invented  by  Mr.  Ged,  Scotland    .        .  1785 


52  FUNDAMENTAL  RULES.  [ART.  III. 

Cotton  first  spun  in  America 1787 

Mesmerism,  or  animal  magnetism,  discovered  by  Mesmer    .  1788 
Sunday  schools  established  in  Yorkshire,  by  Robert  Raikes    1789 
Galvanism,  1767, — its  extraordinary  effects  on  animals  dis 
covered  by  Mrs.  Galvani 1789 

Steam  woollen  factory  first  erected,  at  Leeds  .  .  .  1792 
Flax  spun  by  steam  ........  1793 

Vaccination  introduced  by  Jenner 1798 

Ceres  discovered  by  Piazzi 1801 

Pallas  discovered  by  Gibers  1801 

Life-boats  invented  1802 

Juno  discovered  by  Harding 1804 

Vesta  discovered  by  Olbers 1807 

Steam  first  used  to  propel  boats  by  Fulton,  in  America  .  1807 
Engraving  on  steel  first  invented  by  Perkins,  an  American  1818 
Gas  first  used  for  lighting  streets  in  the  U.  S.,  at  Baltimore  1821 
Egyptian  hieroglyphics  first  decyphered  by  Champollion  1822 
Macadamizing  streets  commenced  in  London  by  McAdam  1824 

First  locomotive  at  Liverpool 1829 

Electro-magnetic  Telegraph  invented  by  Morse,  America  .  1832 
Daguerreotype  impressions  first  taken  by  Daguerre  in  France  1839 
The  existence  of  the  planet  Neptune  predicted  by  Adams  and 

Leverrier 1846 

Magnetic  pendulum  used  for  measuring  longitude   .         .       1848 

21-30.  How  many  years  elapsed  between  the  invention 
of  the  lyre  and  the  invention  of  musical  notes  ?  The  earli 
est  known  astronomical  observations,  and  the  invention  of 
the  signs  of  the  Zodiac  ?  The  first  eclipse  on  record,  and 
the  invention  of  the  telescope  ?  The  use  of  the  mariners' 
compass  in  China,  and  the  introduction  of  solar  quadrants  ? 
The  arrival  of  the  first  ship  in  Greece,  and  the  building  of 
the  first  steamboat  ?  The  establishment  of  the  first  public 
library  at  Athens,  and  the  invention  of  printing  ?  The  in 
vention  of  the  hour-glass,  and  the  invention  of  watches  ? 
The  use  of  paper  in  China,  and  the  manufacture  of  paper 
out  of  linen  ?  The  first  manufacture  of  glass,  and  the  in 
vention  of  spectacles  ?  The  discovery  of  the  Vulgate  edition 
of  the  Bible,  and  the  establishment  of  Sunday  schools  ? 

31-40.  Allow  3651  days  for  each  year,  and  reduce  to 
weeks  the  number  of  years  that  elapsed  between  the  manu- 


§8.]  TRACTIONS   AND   COMPOUND   NUMBERS.  53 

facture  of  porcelain  in  China,  and  the  manufacture  of  glass 
in  England  ?  The  invention  of  pendulum  clocks,  and  the 
use  of  the  magnetic  pendulum  for  measuring  longitude  ?  The 
introduction  of  figures  by  the  Arabs,  and  the  invention  of 
decimal  arithmetic  ?  The  invention  of  the  steam  engine, 
and  the  introduction  of  locomotives?  The  invention  of 
wind-mills,  and  the  application  of  steam  to  grist-mills  ?  The 
first  astronomical  observations  known  to  have  been  made  at 
Babylon,  and  the  prediction  of  the  planet  Neptune  ?  The 
invention  of  the  sun-dial,  and  the  invention  of  watches  ?  The 
use  of  the  compass  in  China,  and  the  discovery  of  the  varia 
tion  of  the  needle  ?  The  introduction  of  the  art  of  painting 
into  Rome,  and  the  invention  of  the  daguerreotype  ?  The 
regulation  of  the  Julian  year,  and  the  introduction  of  the 
new  style  into  England  ? 

41-43.  Give  the  answer  to  each  of  the  remaining  ques* 
tions  on  this  page,  in  years,  months,  and  days, — in  years 
and  fractions  of  a  year, — and  in  years  and  decimals  of  a 
year.  What  length  of  time  elapsed  between  the  birth  of 
John  Calvin,  July  10,  1509,  and  the  birth  of  Oliver  Crom* 
well,  April  25,  1599  ? 

44-52.  What  length  of  time  elapsed  between  the  birtb 
of  Fenelon,  August  6,  1651,  and  the  birth  of  William 
Penn,  October  14,  1644  ?  Between  the  birth  of  Galileo* 
Galilei,  February  19,  1564,  and  the  birth  of  Sir  William 
Herschel,  November  15,  1738  ?  Between  the  birth  of  Sir 
Francis  Bacon,  January  22,  1561,  and  the  birth  of  Benja* 
min  Franklin,  January  17,  1706  ? 

53-67.  What  length  of  time  elapsed  between  the  birth 
of  Tycho  Brahe,  December  19,  1546,  and  the  birth  of  Sir 
Isaac  Newton,  December  25,  1642  ?  The  birth  of  George 
Washington,  February  22,  1732,  and  Patrick  Henryr  May 
29,  1736?  The  birth  of  William  Shakspeare,  April  23, 
1564,  and  John  Milton,  December  9,  1608  ?  The  birth 
of  John  Adams,  October  19,  1735,  and  Thomas  Jefferson, 


54  FUNDAMENTAL  RULES.  [ART.  III. 

April  2,  1743  ?    The  birth  of  Roger  Sherman,  April  19, 
1721,  and  Benjamin  West,  October  10,  1738  ?a 

68.  How  many  days  have  elapsed  since  the  birth  of  Sir 
Humphrey  Davy,  December  17,  1779  ? 

69.  The  equatorial  diameter  of  the  earth  is  7970  miles, 
and  the  circumference  is  3|  times  the  diameter.     If  a  man 
6  feet  high  were  to  travel  round  the  earth,  how  many  yards 
farther  would  his  head  travel  than  his  feet  ?b 

70.  A  pound  Avoirdupois  =  14oz.   lldwt.   16gr.  Troy. 
What  part  of  a  Troy  ounce  is  an  ounce  Avoirdupois  ? 

Ans.  jjf . 

71-75.  Find  the  difference  of  latitude  and  longitude0 
between  Boston  and  Philadelphia.  New  York  and  St. 
Louis.  Charleston  and  New  Orleans.  Cape  Horn  and  the 
Cape  of  G-ood  Hope.  Paris  and  St.  Petersburg. 

76.  Assuming  1  ton  as  the  average  amount  of  carbonic  acid 
produced  by  6100  persons  in  24  hours,  and  estimating  the 
total  population  of  the  globe  at  759999000,  how  many  tons 
would  be  produced  daily  by  human  respiration  ? 

Ans.    124590  tons. 

77.  The  average  length  of  the  tropical  year  is  365d.  5h. 
48m.  49T7Q  sec.d     How  many  days  in  4  centuries,  and  what 
fraction  of  another  day  ? 

78.  A  balance  made  by  Ramsden,  for  the  Royal  Society, 
is  capable  of  weighing  10  pounds  avoirdupois,  and  turns 
with  .01  of  a  grain.6     What  part  of  the  weight  is  required 
to  turn  the  scale  ? 


*  Encyclopaedia  Americana,  Rees'  Encyclopaedia,  Belknap,  Biog. 
American,  Allen. 

t  Keith. 

c  See  Table,  $  9. 

d  Somerville.  The  tropical  or  civil  year  is  the  time  that  elapses 
between  two  consecutive  returns  of  the  sun  to  the  same  equinox  or 
solstice. 

e  Ure's  Dictionary. 


§9.] 


LATITUDES   AND   LONGITUDES. 


55 


9.  TABLE  OF  LATITUDES  AND  LONGITUDES, 

AND  DISTANCES  FROM  WASHINGTON,  OF  THE  PRINCIPAL 
CITIES,  OBSERVATORIES,  NAVAL  STATIONS,  ETC. 

FROM  THE  UNITED  STATES  AND  AMERICAN  ALMANACS. 


Place.                         State.                         Object. 

Longitude 
from 
Greenwich. 

Latitude. 

Dist. 

from 
Wash 

o    '     // 
W.   2    542 
E.  22  17  12 
W.  73  42  49 
"    77    645 
E.     9  56  39 
W.  72  31  28 
"    7633 
"    85    5  15 
"      6  38  53 
"    7628 
"    81  54 
"    6950 
"   704737 
"    7637 
"    68  47 
"    70  18  34 
"    78  13 
«    804123 
"         28    0 
E.  132353 
"             41 
"    112053 
"      7    645 
W.71    4  20 
E.     8  48  59 
"    17    2  30 
W.  73  11  46 
•«    71  17  19 
"    74    0    3 
«    6955 
E.     4  22    8 
"    19    311 
W.7855 
'    7452    6 
'    73  10 
'         2014 
<    67  12  15 
'    71    721 
E.          5  51 
W.8033 
"    77  17 
"    70  34  44 
"    70  11    6 
"    70    4    9 
"    67  16    8 
E     18  28  45 

o     ;     // 
57    857.8 
602656.8 
4239    3 
3849 
53  32  45.3 
42  22  15.6 
38  58  35 
29  37  25 
54  21  12.7 
4255 
3328 
44  18  43 
42  32  11 
39  17  55 
44  47  50 
4142    6 
42  59 
32  25  57 
52    8  27.6 
52  31  13.5 
51  28    2 
44  29  54 
5044    9.1 
42  21  23 
53    4  36 
51    630 
41  10  30 
4140    3 
40  42    0 
4353 
50  51  10.8 
47  29  12.2 
4253 
40    452 
4427 
51  37  44  .3 
45  11  18 
42  22  15 
52  1251.8 
34  17 
4254 
42  38  18 
4333   6 
42    222 
555840s. 
3356    3s. 
44  22  45 
32  46  33 
4222 
38    2    3 
42    0 
59  54  42  4 
39    554 
3357 
3957 
43  12  29 
41    016 

Miles 

370 

7 

383 
40 

883 

333 
575 
595 
452 
40 
663 
466 
374 
635 

440 

284 
409 
226 
570 

381 
156 
513 

786 
437 

473 
341 
470 

507 

671 
540 
441 
121 
717 

492 
506 
393 

481 

Abo                   Finland  Obs  

Albany  N.  Y  Capitol  
Alexandria       D   C  

Amherst  Mass  College  Ch.  .. 

Apalachicola  Bay  .  .  Fa  Light  

Augusta  Me  State  House. 
Baker's  Island  Mass  Light  
Baltimore       •••         Md    St  Mary's  C 

Bangor  Me  State  House.  . 
Barnstable             .     Mass    New  C   H  . 

Beaufort             •    .     S  C  Arsenal    

Bedford  England  Obs  
Berlin  Prussia  Obs.  Old  
BLickheath                 England           Obs     

Breslau         Germany  Obs  

Bridgeport  Conn  Baptist  Ch. 
Bristol     R.I....  Episcopal  Ch. 

Brunswick  Me  College  

Buffalo  N.  Y  
Burlington  N.  J  Obs  

Burlington                  Vt 

Buahy  Heath  England  Obs  

Calais                          Me 

Cambridge  Mass  Obs.  Old  

Camden0  S  C     

Cape  Ann  Mass  N.  Light  
Cape  Elizabeth.    .  .  .  Me  Li^ht  

Cape  Cod  Me  Light  

Cape  of  Good  Hope.  Africa  Obs      

Castine  Me  Fort  
Charleston                  S  C                   St  M   Ch 

W.  68  59  33 
"    795727 
"    71    333 
"    78  31  30 
"   87  30  30 
E.  104457 
W.8424 
«    81    7 
"   83    3 
11    71  29 
E.  28  59 

Charlestown  Mass  Navy  Yard  .. 
Charlottesville  Va  Univ.  Obs..  .  . 
Chicago   Ill    

Christiania  Sweden  Obs  
Cincinnati  Ohio  Obs  
Columbia    S  C      .... 

Columbus  Ohio  
Concord  N.  H  State  Hrmse.. 
Constantinople  Turkey  —  .  .  M  .  St  .  Sophia 

5G 


FUNDAMENTAL   RULES. 


[ART.  in. 


Place.                         State.                         Object. 

Longitude 
from 
Greenwich. 

latitude. 

lYtst. 
from 
Vash 

Copenhagen             .  .Denmark  Obs  

o     '     // 
E.   123457 
W.94  19  15 
"    90  15  15 
E.   195747 
W.84  11 
"    71  1059 
"    82  58 
"    71    4  30 
E.  264345 
W.7054 
"    7530 
"81  29 
"     62030 
"      1  34  30 
"   76    8 
"    6656 

0       /         '/ 

55  40  53 
44    052 
38  37  28 
50    350 
3944 
42  14  57 
42  24 
42  19  10 
58  22  47.1 
4313 
3910 
40  30  52 
53  23  13 
54  46  14.9 
384610 
4454 
36    5 
55  57  23.2 
42  58 
43  46  40.8 
3814 
3834 
46    3 
39  24 
29  15 
40    240 
46  11  59.4 
3855 
33  21 
42  36  44 
50  56    5.2 
51  31  47.9 
42  35  16 
51  28  39 
3937 
44  39  20 
4417 
53  33    5 
4016 
414559 
23    9  26 
40    1  12 
60    942.3 
41  27  15 
42  14 
41  1442.6 
3436 
3955 
4241    8 
32  23 
3836 
55  47  30 
51  30  12.7 
24  32  32 
44    8 
3559 
54  42  50.4 
48    3  24 
40    236 
38    6 
52    928.2 
34  40 
4311 

liles 

461 
422 
524 

438 

490 
120 
372 

80 
769 
274 

474 

542 
56 

43 
144 

2 
4S8 
462 

396 

68 
936 
5'J3 

no 

335 
130 

457 
341 
335 
7(18 
571 
462 
1010 
936 

456 

498 

111 

522 

1065 
403 

Coteau  des  Prairies.  Iowa  Red  Quarry.. 

Cracow..  ~  Poland  Obs  

Dedhnm  Mass  1st  Cong.  Ch. 
Detroit  Mich  

Dorchester  Mass  Obs  
Dorpat  ...    Russia  Obs  
Dover             N   H  

Durham  Scotland  Obs  

Fdenton            •           N    C 

Fdinbur^h  Scotland  Obs 

"      3  10  54 
"    70  55 
E.   11  1554 
W.84  40 

Fxeter  °     .-              N   H  .   .  . 

Florence  Italy  
Frankfort    Ky  

"    66  45 
"    77  18 
"    944715 
"    75  10    9 
E.     6    922 
W.77    5  15 
"    7917 
•<    704019 
E.   1044    6 
"      95638 
W.72  36  32 
0 
"    7735 

Frederick            ..       Md 

Galveston  Texas  

Geneva  Switzerland.  .  Obs  
Georgetown  D.  C  Obs  
Georgetown  S.  C  

Gotha  Germany  
Gottingen  Hanover  Obs  
Greenfield  Mass  2d  Cong.  Ch.. 

Halifax                        N   S 

Hillowell                    Me 

«    69  52  30 
E      9  58  3° 

Harrisburg  Pa  
Hartford  Conn  State  House.. 
Havana  W   I     Moro  Castle 

W.  76  50 
"    72  40  45 
"    82  22  21 
"    751836 
E.  24  57  53 
W.  70  36  38 
"    7346 
"    81  24  54 
"    86  57 
"    86    5 
«    70  46  17 
"    90    8 
"    92    8 
E.  49    638 
W.      11  43 
"    81  48  30 
«    7640 
"   8354 
E.  20  30    8 
"    14    8    9 
W.  76  20  33 
"    84  18 
E.     43153 
W.9212 
"    78  46 

Helsingfors  Finnland  ...   Obs  
Holrnes's  Hole  Mass  Windmill  .  .  ... 
Hudson  NY       

Hudson  Ohio  Obs  
Huntsville  ,        Ala    

Kasan  Russia  Obs  
Kensington  England  Obs  
Key  West  Light  
Kingston                       U    C 

Knoxville  Tenn  

Kremsmunster  Germany  Obs  

Leyden  Holland  Obs  

Little  Rock  Ark   

Lockport  N  .  Y  ».  

§9.] 


LATITUDES  AND  LONGITUDES. 


57 


Place.                        State.                        Object. 

Longitude 
from 
Greenwich. 

Latitude. 

Dist. 

from 
Wash 

Miles 

London  England  St.  Paul's.... 
Lowell  Muss  ...St.  Ann's  Ch.. 

"             545 
"    71  18  57 

50  30  49 

42  38  48 

444 

198 

441 

Madras  India  Ohs  
Manheim  Germany  Ohs  

E.  80  15  57 
"      8  27  51 

13    4    9.2 
49  °9  13  7 

M:irl)lehend  Mass  Light  
Marseilles  France  Ohs  
Matinicus  Rock  Me  Light  
Mexico  Mexico  
Middletown  Conn  W.  Univ  

W.  70  50  39 
"      5  22  15 
"    68  57    ] 
"    99    5 
"    7-2  39 

42  30  14 
43  17  49 
43  46  42 
19  25  45 
41  33    8 

458 
326 

Milan°  Italy  Ohs  
Mobile  Ala  

E.     9  11  48 
W  88  11 

4528    0.7 

1013 

Mobile  Point  Light  
Modem.  Ohs 

"    88    0  36 

301338 

40  45  co 

Montpelier  Vt  

u    70  3(j 

44  17 

516 

E     11  36  3Q 

48    8  45 

Naritucket  Mnss  South  Tower 
Naples  Italy  Ohs  
Nashville  Tenn  Univ.  Obs..  .. 
Natchez  Miss  Castle  
Newark  N  J 

W./O    6  1:2 
E.   14  15    5 
W.8649    3 
«    91  21  42 

41  16  50 
4051  46.6 
36    933 
31  33  48 

490 

684 
1110 

"    70  55  49 

41  38    7 

434 

Newborn  N  C 

"    77    5 

35  °0 

348 

"74     1 

41  31 

°86 

Newburyport  Mass  Light  
Newcastle  Del  

«    704930 
"    75  3%) 

42  48  23 
39  40 

478 
105 

New  Haven  -  Conn  Obs  
Now  London  Conn  -  .  .  .  
New  Orleans  La  N.  E.  Light.. 

"    72  57  30 
"   72    9 
«   89    1  50 
"    71  ''1  14 

41  18  30 
4122 
29    832 
41  28  20 

300 
353 
1172 

408 

"    91    7  15 

38  33  58 

New  York  N.  Y  City  Hall  .... 
Nicolneff  Russia  Obs  

Norfolk  Va  .  •  Farm  Bank 

"    74    1    6 
E.   315847 
W  76  18  47 

40  42  35 
46  58  20.6 

225 
°30 

"    72  38  21 

42  19    8 

380 

Norwich  Conn  

"    72    7 

41  33 

357 

Ormskirk  England  .Obs  
Oxford  .  -  England  Obs 

«      254 
<(      i  15  23 

53  34  18 
51  45  40 

Padua  -Italy  Obs  
Palermo  Italv  Obs.  Old  
Paramatta  N.  S.  Wales  .  Obs  

E.   11  52  18 
"    13  21  24 
"151    135 
"      2  °0  2'J 

4524    2.5 
38    644 
334850s. 
48  50  13 

Pensacola  Fa  .  .  .  .Navy  Yard 

W  87  15  21 

30  20  30 

1050 

Petersburg  Va  ...... 

"    77  20 

37  13  54 

140 

Philadelphia  Pa  State  House 

"    75    945 

39  56  58 

138 

Pittsburo-  Pa 

<«    79  58 

40  26  15 

226 

Pittsfield  Mass  ~lst  Cong.  Ch. 
Plattsburo-  N.  Y  .  ... 

«    73  15  36 
"    73  26 

42  26  55 
44  42 

380 
539 

Plymouth  Mass  Court  House  . 
Portland  „.-.  Me  Tn.  House.  .. 
Portsmouth  N.  H  Unit.  Church  . 
Poughkeepsie  ..N.  Y  
Prague  Bohemia  Obs  
Princeton  N.  J  Nassau  Hall.. 
Providence  R.  I  Univ.  Hall... 
Quebec  L.  C  Citadel  

«    70  40  28 
«    70  20  30 
"    704550 
«    7355 
E.   142529 
W  74  39  33 
«    71  24  48 
«    71  16 

41  5728 
43  39  26 
43    4a5 
4141 
50    5  18.5 
402041 
41  49  32 
46  49  12 

447 
545 
493 
301 

177 

400 

781 

58 


FUNDAMENTAL   RULES. 


[ART.  in. 


Place.                         State.                          Object. 

Longitude 
Green  wkh. 

Latitude. 

Dist. 
from 
Wash 

Rilei^h          N   C 

O       >       ll 

o     /      '; 

Miles 

Regent's  Park  London  Obs  
Richmond    .    .  .-.         Va   -     ..Capitol 

W.7848 
"           91~ 

3547 
513130 

2»8 

Rochester  -  N.  Y  Roch.  House. 
Rome  Italy  Obs  

"    77  27  28 
"    7751 
E.   122841 

37  32  17 
43    8  17 
41  53  54 

117 
369 

Siickett's  Harbour  .  .  N.  Y  _„ 
Saco              Me        

W  .81  52 
"    7557 

40  16  40 
4355 

338 
415 

'  '    70  26 

43  31 

528 

St.  Croix  W.I  Obs  
St.  Fernando  -.Spain  Obs  
St.  Helena    ^.-Africa  Obs  
St.  Joseph's  Bay  Fa  Light  
St.  Louis  Mo  Cathedral  
St   Mark's  Fa  

u   81  35 
"   6441  15 
"     6  12  17 
"     54230 
"    8523  15 
«    90  15  10 

17  44  32 
36  27  43 

15  55  26  s 
2952 

383728 

808 
854 

St.  Petersburg  .Russia  Pulkova  Obs 
S;tlem  .Mass  E.  I.  M.  Hall 
Smdwich.  .........  Mass  1st  Cong.  Ch. 

"   84  11    0 
E.  301946 
W.  70  53  57 
"   703013 

59  46  18.7 
42  31  19 
414531 

448 
456 

Schenectady       .  .       NY 

Slough  England  Obs  
South  Bend  L.  Michigan  -  
Southwick  Mass  Obs  
Speyer  Germany  Obs  
Sp  ringfield  Mass  Court  House  . 
Sprino-field  Ill    ..„. 

"    73  55 
"         36    0 
"   871939 
"    724857 
E.     82638 
W.  72  35  47 

51  30  20 
4137    6 
42    047 
49  18  55.2 
42    6    1 

359 
363 

Stockholm  Sweden  Obs  

E.   18    330 

59  20  31 

Tallahassee  Fa     ..   ..         

Tampa  Bay  Fa  Egmont  Key. 
Ti  Tanka  Tarn.  Lake  Iowa  

£     82  45  15 

2736    4 

Toronto         .-.         .  .U   C 

Tortugas  .  -  .-  Fa  Light  
Trenton                       N  J                            . 

'     82  52  22 

24  37  20 

Troy  N.Y    „ 

40  44. 

Turin  Italy  

45    4    6 

Turtle  Island  Lake  Erie  - 
Tuscaloosa  Ala  Obs  
University  of  Va.  .  .  .  Va  

W.  83  23  35 

"    8742 

4145    9 
33  12 
38    2    3 

858 
l'J4 

Upsala  Sweden  Obs  

Utica       ....                NY                 Dutch  Church 

E.   173842 

59  51  50 

Vandalia            ...       Ill    

W  89    2 

38  50 

781 

Vevay  -  .  .  .   Ind  .   .. 

U      01    cQ 

38  46 

544 

Vienna  Austria  Obs  
Vincennes      Ind  

E.  162259 
W  87  25 

48  12  35.5 
38  43 

688 

WASHINGTON  D.  C  Capitol  

"    77    1  30 

38  53  23 

o 

Washington    Miss  _  _ 

"    91  20 

31  36 

104 

E    20  58  23 

52  13    1 

Weasel  Mountain.  .  .N.  J  C.  S.  St.  .  . 
West  Hills  N.Y  C.  S.  St  
West  Quoddy  Head  Me  Light  
Wheelin^  Va  

W.7412  15 
"    73  25  17 
"    66  57  19 
"   80  42 

40  52  35 
404849 
1449    4 
40    7 

260 
264 

Willi  unstown  Mass  C.  Church  ...  . 
Wilmington  Del  

'    731320 

'    75  28 

42  42  51 
39  41 

396 
110 

Wilmington  N   C  -. 

'    78  10 

34  11 

405 

Wilna  Poland  Obs  
Worcester  Mass  Antiq.  Hall 
York  Me  .   . 

E.  251759 

V.  71  48  10 
<    70  40 

54  41    0 
42  16  13 
3  10 

398 
502 

York  pa 

i    76  40 

39  58 

90 

Yorktown  Va  ..  

"    7634 

3713 

§  10.]  MISCELLANEOUS   EXAMPLES.  59 

1O.  MISCELLANEOUS  EXAMPLES  IN  INTEGERS,  DECI 
MALS,  FRACTIONS,  AND  COMPOUND  NUMBERS. 

1-4.  Fifteen  degrees  of  difference  in  longitude  are  equiv 
alent  to  an  hour's  difference  of  time.  What  difference  of 
longitude  would  make  a  difference  of  four  minutes  in  the 
time  ?  A  difference  of  one  minute  ?  Of  four  seconds  ?  Of 
one  second  ? 

5-6.  Give  a  rule  in  your  own  words  for  determining 
the  difference  of  time  when  the  difference  of  longitude  is 
known.  For  the  difference  of  longitude  when  the  differ 
ence  of  time  is  known. 

7-10.  What  is  the  difference  of  longitude  between  two 
places,  if  the  difference  of  time  is  2h.  15m.  27sec.  ?  If  the 
difference  is  4h.  Om.  16sec.  ?  39m.  45sec.  ?  lOh.  48m. 
49sec.  ? 

11-15.  What  is  the  difference  of  time  between  Boston 
and  Philadelphia  ?  New  York  and  St.  Louis  ?  London 
and  St.  Petersburg  ?  Paris  and  New  Orleans  ?  Washing 
ton  and  Stockholm  ? 

16-20.  When  it  is  noon  at  Cincinnati,  what  time  is  it  at 
Charleston  ?  At  Little  Rock  ?  At  Florence  ?  At  Jeffer 
son  ?  At  Raleigh  ? 

21-25.  When  it  is  Ih.  20m.  25sec.  P.  M.  at  Providence, 
what  time  is  it  at  Paramatta  ?  At  Vera  Cruz  ?  At 
Albany  ?  At  Harrisburg  ?  At  Hartford  ?* 

26.  Estimating  the  pound  sterling  at  $4.84,  in  how  many 
years  would  the  agricultural  products  of  the  United  States 
pay  the  British  National  Debt,  the  annual  amount  of  the 
former  being  about  $252,240,779,b  and  the  latter  amounting 
in  1847  to  £764,608,284  ?c 

*  The  table  in  $9,  furnishes  materials  for  an  indefinite  number  of 
questions,  similar  to  these. 

b  Patent  Office  Report,  1847.  «  Annual  Register. 


60  FUNDAMEOTJLL   RULES. 

27.  French  Invoice:— 


[ART.  in. 


Lyon,  9  Avril,  1848. 

Messrs,  Merrit  &  Lamb  De»ivt. 

a  Perigord  &  Lubin. 

No,  40,  Une  caisse  par  roulage  et  paquebot  Bavaria 
PL  du  24  Avril. 


568a 

Oavates  longues 

6  J.i  .'42 

« 

290 

50 

569 

"        f  ecossais 

11T82 

36 

" 

570 

44                  u 

20  j£ 

42 

a 

571 

44                         44 

8T92 

572 

44                         41 

6T82 

573 

41                         U 

5_42 

574 

44                         44 

81 

28T92 

48 

a 

575 

tc                  <( 

19 

54 

« 

576 

U                          44 

5 

48 

" 

577 

44                          4t 

•2_62 

36 

<t 

578 

44                          4< 

8 

30 

» 

4530 

EniMlage 

45 

Conim,  3°|0 

137 

!25 

F. 

4T12 

25 

28,  How  many  centuries  in  a  trillion  seconds,  allowing 
95  leap  years  in  each  4  centaries,  and  24  leap  years  in  each 
of  the  remaining  centuries,  and    how  many  yeaisr  days, 
ihours,  minutes,  and  seconds  of  another  century  ? 

29,  If  a  cannon-ball  could  be  fired  from  the  earth  to  the- 
sun,  and  move  uniformly,  in  a  straight  line,  at  the  rate  of 
1600  feet  per  second,  in  what  time  would  it  reach  the  sun, 
the  mean  distance  being  estimated  at  95000000  miles? 

*  Ne,  of  design. 


§io.] 


MISCELLANEOUS   EXAMPLES. 


61 


30.  Invoice  of  Goods  shipped  by  M.  Naylor  &  Son-s^.  of  Lon 
don,  in  the  Westminster,  Warren,  Master,  for  Boston;  by 
order  and  for  account  and  risk  of  Messrs.  W.  Appleton  &  Co. 


Two  Chests  Asafoetida, 
Gr.    5  0     7  tr.      2  24 
"     5  0  24    "       2  24 


Gr.  10  1     3  tr.  1  1  20 
1  1  22  dft.  2 


839 

Disct.  2|  per  cent. 


Cording, 


Two  boxes  Kino, 


Gr. 


2  21  tr.  21 
2  25  "   23 

1  1  18  tr.  44 

1  18  dft.  2 


100 

Disct.  2 1  per  cent. 


Cording, 


One  Cask  Litharge,  M.  N. 
Gr.  6  1  8 
Tr.       1  1 


607 


Cording, 


Charges, 
Customs  entries, 

Cartage,  wharfage,  and  shipping, 
B.  of  Lading  2s.  6d.,  Postages  2s. 


Brokerage  and  com.,     Spercent. 
Insurance  £40. 
Stamp  2s.    Comm.  Is. 


E.  E.     London,  27th  Jaa.,  1850. 
M.  NAYLOR  &  SONS. 


15 


22  6 


11 


14 


.  rf. 


31. 


JUNE, 


FUNDAMENTAL  RULES. 
TIME-BOOK. 


[ART.  III. 


1849. 


NAMES. 

25 

26 

'11 

28 

29 

80 

Total. 

peTJEk. 

Amount. 

REMARKS. 

George  Brooks, 

1 

1 

I 

1 

1 

1 

6 

10.50 

Ex.  workm'n. 

Frank.  Jones, 

a 

* 

1 

1 

1 

1 

4* 

6.00 

Rather  slow. 

John  Smith, 

1 

1 

1 

1 

3 

1 

5* 

5.00 

Intemperate. 

Win.  Brown, 

1 

1 

a 

1 

i 

a 

4 

9.00 

Good  hand. 

Thos.  Martin, 

1 

1 

3 

1 

i 

1 

5 

4.00 

Careless. 

Find  the  amount  of  each  man's  wages,  and  the  total 
amount  of  wages  for  the  week. 

32.  If  150  leaves  of  paper  make  a  pile  an  inch  high,  and 
each  leaf  is  twice  the  thickness  of  a  hair,  what  would  be  the 
extent  of  a  quadrillion  hair-breadths  ? 

33.  In  what  length  of  time  would  light,  which  moves  at 
the  rate  of  192000  miles  per  second,  reach  us  from  a  star 
that  is  one  quintillion  miles  distant  ? 

34-41.  Express  1853  by  a  scale  of  notation  that  shall 
have  9  for  its  base,  instead  of  10. a  Express  the  same  num 
ber  by  a  scale  of  7 ;  of  5;  of3;  of  2;  of  8;  of  6;  of  4. 

42-44.  In  the  first  three  weeks  of  June,  1849,  the  num 
ber  of  gallons  of  water  consumed  in  the  city  of  Philadelphia 
was  as  follows :  June  1st.  6439650 ;  2d.  6163665 ;  3d. 
3955785;  4th.  6041005;  5th.  6102335;  6th.  5887015; 
7th.  4569085;  8th.  5887680;  9th.  5489035;  10th. 

*  If  9  had  been  adopted  as  the  base  of  our  system,  9  units  of  any 
order  would  have  made  one  of  the  next  higher  order.     Therefore  if  we 
wish  to  reduce  14646  to  the  scale  of  9,  we  divide  by  9     9)1464fi 
and  find  that  there  are  1627  units  of  the  2d  order,  and  3 
of  the  1st.     The  1627  units  of  the  2d  order  are  equiva 
lent  to  180  units  of  the  3d  order,  and  7  of  the  2d.     The 
180  units  of  the  3d  order  make  20  units  of  the  4th  order 
and  0  of  the  3d.     The  20  units  of  the  4th  order  make  2 
units  of  the  5th  order  and  2  of  the  4th.     The  whole 
number  would  therefore  be  expressed  thus  :  22073. 


9)1627-3 
9)180-7 


9)2-2 


MISCELLANEOUS   EXAMPLES. 


63 


3557140;  llth.  4753075;  12th.  4395096;  13th.  5703690; 
14th,  5550365;  15th.  6346655;  16th.  6623640;  17th. 
5550365;  18th.  6224995;  19th.  5918345;  20th.  6565310; 
21st.  7239945.  What  was  the  total  consumption  of  the 
three  weeks  ?  The  average  daily  consumption  ?  If  the 
whole  quantity  were  put  into  100-gallon  casks,  and  the 
casks  were  arranged  side  by  side,  how  far  would  they  extend, 
each  cask  occupying  a  space  of  3  feet  ? 

45-50.  Determine  from  the  following  table,  the  area  of  the 
United  States  and  Territories,  the  amount  of  exports  and 
imports,  in  the  years  1840  and  1847,  and  the  number  of 
inhabitants  to  a  square  mile  in  each  state  and  territory,  by 
the  census  of  1840. 


STATES. 

No.  Square 
Miles. 

Exports 
1840. 

Exports 
1847. 

Imports 
1840. 

Imports 
1847. 

Maine 
New  Hampshire 
Vermont 
Massachusetts 
Rhode  Island 
Connecticut 
New  York 
New  Jersey 
Pennsylvania 
Delaware 
Maryland 
Distof  Columbia 
Virginia 
North  Carolina 
South  Carolina 
Georgia 
Alabama 

a35,000 
8,030 
8,000 
7,250 
1,200 
4,750 
46,000 
6,851 
47^000 
2,120 
11.000 
50 
61,352 
45,500 
28,000 
58,000 
50,722 
47  147 

1>$1,018,269 
20,979 
305,150 
10,166,261 
206,989 
518,210 
34,264,080 
16,076 
6,820,145 
37,001 
5,768,768 
753,923 
4,778,220 
387,484 
10,036,769 
6,862,959 
12,854,694 

b$l,634,203 
1,690 
514,298 
11,248,462 
192,369 
599,192 
49,844,368 
19,128 
8,544,391 
235^59 
9,762,244 
124,269 
5,658,374 
284,919 
10,431,517 
5,712,149 
9,054,580 

t>$628,762 
114,647 
404,617 
16,513,858 
274,534 
277,072 
60,440,750 
19,209 
8,464.882 
802 
4,910,746 
119,852 
545,685 
252,532 
2,058,870 
491,428 
574,651 

b$574,056 
16,935 
239,641 
34,477,008 
305,489 
275,823 
84,167,352 
4,837 
9,582,516 
12,722 
4,432,314 
25,049 
386,127 
142,384 
1,580,658 
207,180 
390,161 
336 

Louisiana 
Florida 
Texas 

46,431 
59/268 
305  520 

34,236,936 

1,858,850 

42,051,633 

1,810,538 

10,673,190 
190,728 

9,222,969 
143,298 
29  826 

37  680 

2  241 

26  956 

44  000 

28  938 

1  256 

Ohio 
Indiana 

39,964 
33,809 

991,954 

778,944 

4,915 

90,681 

Illinois 

55  405 

52  100 

266 

Michigan 

56,243 

50  914 

162,229 

93,795 

138,610 

37,603 

Wisconsin 

53,924 

67  380 

10  600 

167  195 

52  198 

Minesota  Terr. 

ItJIi.OOO 

Missouri     " 

579,000 

Oregon       " 

341,463 

Indian         " 

248,851 

New  Mexico 

77,387 

California 

448,691 

*  U.  S.  Land  Office   Documents,  1849 ;  see   also,   12th  Ann.  Rep. 
Mass.  Board  of  Educ.,  pp.  34,  35.  b  Am.  Almanac. 


64  FUNDAMENTAL   RULES.  [ART.  HI. 

51.  Water  is  composed  of  two  volumes  of  hydrogen  and 
one  volume  of  oxygen,  a  volume  of  oxygen  being  of  the 
same  weight  as  16  volumes  of  hydrogen.  Required  the 
weight  of  each  gas  in  a  cubic  foot  of  water,  which  weighs,  at 
the  temperature  of  62°  Fahrenheit,  62.51b. 

52-55.  Give  abbreviated  rules  for  multiplying,  and  for 
dividing*  by  5;  by  25 ;  by  125;  by  625. 

11.  TEST  EXAMPLES. 

1.  From  the  sum  of  287.53  +  195.7  +  6008  +  7.975  + 
3092.06,  subtract  by  a  single  operation  the  sum  of  948  + 
27.008  +  1090.3  +  4726.87  +  95.953.b 

2-10.    Reduce  to  whole  numbers  &ffi-,    4-6g8-8;    2f!; 

1001.    1001.     UHU;     9875.     ^876.     1=^)43. 

11-15.  Reduce  137  to  8ths;  to  4ths;  to  llths;  to 
125ths;  to  1016ths. 

16-25.  Reduce  to  simple  fractions  f  of  7 ;  J  of  4  of  3£ ; 
f  of  3}  of  2ft;  }J  of  A  of  90;  Jf ;  ^;  ^  &;  g|; 

»/  of  1  * ,of  T6f  of  ?i  of  161. 

_____ 85 

a  Since  jys=.04  and  25=.^,  we  may  obtain  ^~  of  a  number, 
(which  is  equivalent  to  dividing  the  number  by  25,)  by  multiplying  by 
.04,  and  we  may  obtain  25  times  a  number,  by  dividing  by  .04.  By 
finding  the  decimal  value  of  J,  &c.,  similar  rules  may  be  formed  for 
each  of  the  other  cases. 

b  The  arithmetical  complement  of  a  number  is  the  difference  between 
the  number  and  some  power  of  ten.     Such  examples  as  the  1st  in  this 
section  can  be  most  readily  solved  by  addition,  taking 
the  arithmetical  complement  mentally,  of  each  of  the  4692.8 

numbers  to  be  subtracted.     This  may  be  done  by  taking         — 271.04 
each  of  the  figures  from  9,  except  the  right  hand  figure,       — 2637. 
which  should  be  taken  from   10.     After  adding  all  the  1384.2 

arithmetical  complements,  we  must  deduct  1  from  the      • 

next  place  at  the  left.     For  example,  if  the  sum  of  4692.8  3168.96 

—  -271.04  —  2637+1384.2,    were   required,    writing    the 
numbers  as  in  the  margin,  we  commence  by  saying  4  from  10  leave  6  ; 
2+9+8  =  19;   1+4+3+8+2  =  18;  1+8+6+2+9=26;  2+3+ 
3+7+6=21;  2+1+7—1+4  =  13;   1—1=0. 


§11.]  TEST   EXAMPLES.  65 

26-30.  Find  the  value  of  15/3  +  3f|-  +  27f  +  f  +  41JJ; 
|  of  3  Jr  -  f  of  -}  |  of  7f  ;  |  of  J»T  of  8f  X  5  J  X  f  of  1  JT  ; 


31-35.  Reduce  to  mixed  numbers  fj;   JT8S7;  \^°; 

4.0  1  75 
186     ' 

36-40.  Reduce  to  a  fractional  form  49f|;  108f  J;  .0932; 
231}f  ;  8.087. 

41-45.  Reduce  to  the  lowest  terms  JJf;    lil;   Iff; 

fi  3  2  .       492 
H36J     2964' 

46-55.  Find  the  numerators  and  denominators  tnat  are 
indicated  by  a  blank  in  the  following  fractions;  J=ii- 

4=a;  ft=*j  *=TT;  4!=^;  8T4r=TI;  53=45  71= 

-J  |of|=^j  2J:6i=2J. 

56-60.  Reduce  to  fractions  or  mixed  numbers  .18  ;  2.05  ; 
.82;  27.854;  .0563. 

61-65.  Reduce  to  decimals  |J;   |;  T42%;   «;  $fc. 

66-70.  Reduce  to  the  least  common  denominator  |-J  and 

i?;  41  and|f5  i.l./i.^i^  A>&>an<Hfj  «.il> 

and  A. 

71.  A  clerk,  in  balancing  his  books,  found  an  error  of 
S407.25.  What  was  the  probable  cause  of  the  error  ?* 

72-75.  Divide  .0072  by  576000;  27.9  by  .00124; 
46500  by  .0002976  ;  87.002  X  1.008  by  963  x  72000. 

76-80.  Reduce  8s.  7d.  2qr.  to  the  decimal  of  a  £;  2yd. 
2ft.  3in.  to  the  fraction  of  a  furlong;  T8T  miles  to  fur.,  r., 
yd.,  ft.,  and  in.  ;  7mo.  21d.  to  the  fraction  of  a  year;  4|d. 
to  seconds. 

a  The  difference  between  any  number,  and  the  same  number  trans 
posed  in  any  way,  is  divisible  by  9.  Thus,  723—327,  723—372,  723— 
273,  723—237,  are  each  exactly  divisible  by  9.  Therefore,  if  we  find 
in  comparing  the  books  of  a  counting-room  or  banking-house,  that 
they  do  not  agree,  and  the  amount  of  their  disagreement  is  divisible  by 
9,  we  know  that  it  may  have  arisen  from  a  transposition.  We  shall 
thus  frequently  be  enabled  to  discover  an  error  readily,  which  would 
otherwise  have  required  a  long  and  tedious  examination. 

5 


66  MEASURES,  WEIGHTS,  ETC.  [ART.  IV. 

81-85.  Multiply  in  a  single  linea  4793  by  27;  by  103; 
by  251 ;  by  19 ;  by  874. 


IV.  MEASURES,  WEIGHTS,  AND  CUR 
RENCIES. 

THE  standard  weights  of  nearly  every  country,  are  de 
rived  from  the  linear  measures.  Coins  are  made  of  platina, 
gold,  silver,  or  copper.  As  gold  and  silver  are  too  soft  to 
be  used  by  themselves,  some  other  metal  is  mixed  with  them 
before  coining.  The  metal  which  is  added  is  called  alloy. 

12.     STANDARDS  OF  THE  UNITED  STATES.5 

Congress  has  never  fully  exercised  the  power  granted  to 
it  by  the  constitution,  of  establishing  a  uniform  standard  of 
weights  and  measures.  The  standards  used  at  all  the  Cus- 
tom-Houses  were  prepared  by  Mr.  F.  R.  Hassler,  in  1835-6, 


*  Multiplication  in 
cise,  and  in  many 
cases  it  is   much 
more    expeditious 
than   the  ordinary 
method.     The  ac 
companying  exam 
ple,  of  the  multi 
plication   of   4872 
by  3956,  will  show 
the    several    pro- 

a  single  line  will  be  found  a  very  valuable  exer- 

4872 
3956 

6X4+6X8+6X7+6X2 

9X4+9X8+9X7+9X2 
3X4+3X8+3X7+3X2 

12  +  60  +  113  +  133  +  101  +  52+12 
ra.       h.th.     t.th.         th.            h.         tens       un. 

ducts  that  are  to  be  added  to  obtain  each  figure  of  the  entire  product, 
and  will  perhaps  render  the  process  as  intelligible  as  it  could  be  made 
by  any  formal  rule.    To  obtain  the  product  in  a  single  line, 
we  say  6X2=  12.   Set  down  2  and  carry  1.    1+6  X  7+5 X  4872 

2  =  53.     Set  down  3  and  carry  5.     5+6x8+5X7+9x2 
=  106.     Set  down  6  and  carry  10.     10+6x4+5x8+9      19273632 
X7+3x2  =  143.     Set  down  3  and  carry  14.     14+5x4 
+9X8+3X7=127.     Set  down  7  and  carry  12.     12+9x4+3x8 
=  72.    Set  down  2  and  carry  7.     7  and  3X4=  19,  which  we  set  down, 
making  the  entire  product  19273632. 
b  A.  D.  Bache.     Report  on  Weights,  Measures,  and  Balances. 


§  12.]  STANDARDS   OF  THE   UNITED   STATES.  67 

and  are  similar  to  those  used  in  England,  anterior  to  the 
passage  of  the  "Act  of  Uniformity/7  in  May,  1834. 

Many  of  the  states  have  attempted  to  establish  uniformity 
within  their  own  limits,  and  have  passed  laws  for  that  pur 
pose.  There  is,  therefore,  a  slight  diversity  in  the  usages 
of  different  sections  of  the  Union,  but,  as  nearly  all  the  laws 
have  assumed  the  English  system  for  their  basis,  it  does 
not  seem  desirable  to  attempt  making  an  abstract  of  them. 
The  teacher,  however,  should  make  his  pupils  familiar  with 
the  laws  that  have  been  passed  on  the  subject  by  the  legis 
latures  of  their  own  state. 

There  is  a  great  discrepancy  in  the  statements  of  differ 
ent  writers  on  arithmetic,  relative  to  the  government  stand 
ard.  Many  of  those  who  have  alluded  to  the  subject, 
seem  to  have  regarded  the  local  customs  of  their  own 
neighborhood  as  identical  with  the  practice  of  the  United 
States'  officers,  and  probably  no  one  has  given  correct  infor 
mation  as  to  the  standards  in  actual  use  at  the  Mint,  the 
Custom-Houses,  and  in  all  the  departments  of  the  General 
Government. 

In  the  joint  resolution  of  June  14,  1836,  the  Secretary 
of  the  Treasury  is  "  directed  to  cause  a  complete  set  of  all 
the  weights  and  measures  adopted  as  standards,  and  now 
either  made,  or  in  the  progress  of  manufacture,  for  the  use 
of  the  several  Custom-Houses,  and  for  other  purposes,  to  be 
delivered  to  the  governor  of  each  state  in  the  Union,  or 
such  person  as  he  may  appoint,  for  the  use  of  the  states 
respectively,  to  the  end  that  a  uniform  standard  of  weights 
and  measures  may  be  established  throughout  the  United 
States."  In  order  further  to  secure  this  uniformity,  Con 
gress  directed,  in  1838,  the  preparation  and  distribution  to 
the  states,  of  balances  for  adjusting  weights  and  capacity 
measures.  In  1848,  twenty-one  of  the  states  had  received 
these  standards,  and  a  sufficient  number  had  been  prepared 
to  meet  the  demand  from  the  remaining  states. 


68  MEASURES,  WEIGHTS,  ETC.  [ART.  IV. 

But  in  these  very  standards,  there  is  a  great  want  of  sys~ 
tern.  The  foot  is  subdivided  decimally,  instead  of  being 
divided  into  inches;  the  decimal  multiples  of  the  Troy 
pound,  and  the  decimal  sub-multiples  of  the  Avoirdupois 
pound  are  given,  although  they  are  never  used.  The 
whole  matter  is,  therefore,  in  great  confusion,  not  only  in 
this  country,  but  in  every  other  country,  except  France. 
Some  steps  have  been  taken  towards  the  adoption  of  a  uni 
form  international  standard,  and  it  is  not  improbable  that 
some  modification  of  the  French  system  will  eventually  come 
into  general  use  throughout  the  civilized  world.* 

1.   LONG  MEASURE. 

The  denominations  are  Leagues,  Miles,  Furlongs,  Rods, 
Yards,  Feet,  and  Inches. 

Le.     m.        f.           r.             yd.               ft.  in. 

I  =  3  =  24  =  960  =  5280    =  15840  =  190080 

1  =    8  =  320  =  1760    =    5280  =  63360 

1  =    40  =    220    =      660  =  7920 

1  =       5£  =       16J  =  198 

1    =          3  =  36 

1  =  12 

The  English  standard  unit  of  Long  Measure  is  the  yard, 
which  is  equivalent  to  ||  J§§§  of  the  length  of  a  " pendulum 
vibrating  seconds  of  mean  time  in  the  latitude  of  London, 
in  a  vacuum  at  the  level  of  the  sea."b  The  United  States 
standard,  the  original,  of  which  the  state  standards  are 
copies,  is  a  brass  scale  of  82  inches  in  length,  prepared  for 
the  survey  of  the  coast  of  the  United  States,  by  Troughton, 
of  London,  and  deposited  in  the  Office  of  Weights  and  Mea 
sures. 

The  Rod  is  sometimes  called  Perch,  or  Pole. 

The  Yard,  for  CLOTH  MEASURE,  is  subdivided  into  Quar 
ters  and  Nails. 

»  Prof.  McCulloh,  U.  S.  Mint.  b  McCulloch. 


§  12.]  STANDARDS   OF  THE   UNITED    STATES.  69 

yd.       qr.       na.       in. 

1  =4  =  16  =  36 

1=4=9 

1=    2J 

Surveyors  use  CHAIN  MEASURE,  in  which  the  unit  is  a 
Chain  of  4  Rods.  It  is  subdivided  into  Poles  and  Links. 

Mile.    fur.      ch.       poles.          I.  in. 

1  =  8  =  80  =  320  =  8000  =  63360 

1  =  10  =    40  =  1000  =    7920 

1  =      4  =    100  =      792 

1  =     25  =      198 

1  =         7.92 

A  lot  of  land  measuring  10  chains  in  length  and  1  in 
breadth,  contains  an  acre. 

A  palm  =  3  inches ;  a  hand  =  4  inches ;  a  span  =  9 
inches ;  a  pace  =  3  feet ;  a  fathom  =  6  feet ;  a  knot,  or 
geographical  mile,  is  fa  of  a  degree,  or  3T|o^  °f  tne  earth's 
circumference,  and  is  equivalent  to  1.15257  statute  miles, 
or  6085.56  feet;1  a  degree  at  the  equator,  is  691  miles. 

The  inch  is  generally  subdivided  on  scales  into  lOths  or 
decimal  parts,  but  sometimes  into  halves,  quarters,  eighths, 
and  sixteenths.  In  the  work  of  carpenters  and  other  me 
chanics,  the  duodecimal  division  is  sometimes  employed, 
The  inch  =  12  lines,  or  primes ;  the  prime  =  12  seconds ; 
the  second  =  12  thirds,  &c.,  &c. 

a  This  is  the  value  of  the  geographical  or  nautical  mile,  employed 
in  the  Topographical  Bureau  at  Washington,  in  1849.*  The  rate  of  a 
ship's  sailing  is  determined  by  the  half-minute  glass  and  log-line.  The 
log  is  a  piece  of  board,  loaded  on  one  end  so  that  it  will  stand  verti 
cally  in  the  water.  The  intervals  between  the  knots  on  the  line,  are 
intended  to  bear  the  same  proportion  to  a  sea  mile,  as  a  half  minute  to 
an  hour.  Thus  if  8  knots  on  the  line  run  off  of  the  reel,  while  the 
sand  is  running  out  of  the  half-minute  glass,  the  vessel  is  moving  8 
knots  an  hour. 

The  length  of  the  knots  on  the  log-line  would  accordingly  be  y^ 
of  a  nautical  mile  or  50.713  ft.,  if  perfect  accuracy  were  required.  But 
in  order  to  keep  the  ship  "  behind  her  reckoning,"  and  avoid  the  dan 
ger  of  running  ashore,  they  are  made  three  or  four  feet  shorter.  The 
length  of  a  knot  for  a  28  seconds'  glass  is  usually  6£  fathoms. 

*  Determined,  for  the  Bureau,  by  John  Downes,  from  Bessel's  Ele 
ments  of  the  Terrestrial  Spheroid. 


70  MEASURES,  WEIGHTS,  ETC.  [ART.  IV. 

2.  SQUARE  MEASURE. 

The  Square  Mile  is  subdivided  into  Acres,  Roods,  Rods, 
Yards,  Feet,  and  Inches. 

M.      A.  R.  sq.r.  sq.yd.  sq.ft.  sq.  in. 

1=  640  =  2560  =  102400  =  3097600  =  27878400  =  4014489600 

1  =        4  =        160  =        4840  =        43560  =        6272640 

1=          40=        1210=        10890=        1568160 

1  =(5£)*  30£  =(16£)2  272  J=  (198)2  39204 

1  =  (S)2         9  =  (36)  2    1296 

1  =  (12,2       144 

A  square  piece  of  land,  measuring  209  feet  (or  nearly  70 
paces),  on  each  side,  is  about  equivalent  to  an  acre. 

The  value  of  the  several  denominations  in  Square  and 
Cubic  Measure,  is  determined  by  the  standards  employed 
in  Long  Measure. 

3.  CUBIC  MEASURE. 

The  denominations  are  Cubic  Yards,  Cubic  Feet,  and  Cu 
bic  Inches. 

c.  yd.         e.ft.  e.in. 

1  =  (3)3  or  27  =  (36)3  or  46656 
1  =  (12)3  or    1728 

A  foot  of  wood  is  16  cubic  feet.  8  feet  of  wood,  or  128 
cubic  feet,  make  a  cord.  A  ton  of  timber,  storage  or  ship 
ping,  is  40  cubic  feet.  A  perch  of  stone  is  24  f  cubic  feet 
=  1  perch  square  and  1£  feet  thick.  A  square  of  earth  is 
a  cube  measuring  6  feet  on  each  side,  and  is  equivalent  to 
216  cubic  feet.  In  measuring  round  timber,  a  deduction  is 
sometimes  made  from  the  diameter  of  each  stick,  to  allow 
for  waste  in  sawing.  Therefore,  a  ton  of  round  timber, 
although  nominally  but  40  feet,  often  contains  about  50 
feet.* 

*  In  England  40  c.  ft.  of  round  timber,  or  50  c.  ft.  of  hewn  timber, 
make  1  load  or  ton.  In  the  American  lumber  yards,  the  custom  is 
nearly  or  quite  universal  of  estimating  the  ton  at  40  ft.  for  both  hewn 
and  round  timber.  But  as  there  are  different  modes  of  measurement 


§12.]  STANDARDS    OF   THE   UNITED    STATES.  71 

Boards  are  measured  by  the  superficial  foot ;  but  if  they 
are  more  than  1  inch  thick,  allowance  is  made  for  the  addi 
tional  thickness.  A  board  1J  ft.  wide,  20  ft.  long,  and  of 
any  thickness  not  exceeding  1  inch,  contains  20  X  1  J  =  30  ft. 
But  if  it  is  H  inches  thick,  it  will  contain  20  X  1J  X  H 
=  371  ft. 

4.  LIQUID   MEASURE. 

The  denominations  are  Hogsheads,  Gallons,  Quarts, 
Pints,  and  Gills. 

hhd.    gall.        qt.  pt.  gi. 

1  =  63  =  252  =  504  =  2016 

1  =      4  =      8  =      32 

1=      2  =        8 

1  =        4 

The  United  States  standard  for  measuring  liquids,  is  the 
gallon,  which  is  a  vessel  containing  58372.2  grains  (8.3389 
pounds  avoirdupois)  of  the  standard  pound  of  distilled  water, 
at  the  temperature  of  39°.83  Fahrenheit,  the  vessel  being 
weighed  in  air  in  which  the  barometer  is  30  inches  at  62° 
Fahrenheit.  This  corresponds  very  nearly  with  the  Eng 
lish  wine  gallon,  which  contains  231  cubic  inches.  Milk 
and  malt  liquors  are  sold  by  Beer  Measure  in  many  places, 
the  beer  gallon  containing  282  cubic  inches.  The  hogs 
head  (measure)  is  used  only  in  estimating  the  contents  of 
cisterns,  wells,  or  large  bodies  of  water. 

The  Gallon  (Cong.a)  is  subdivided  by  apothecaries,  into 
Pints  (Ob),  Fluidounces  (fg),  Fluidrachms  (fe),  and  Min 
ims  (m). 

Cong.    O.        f^  f£  HI 

1  =  8  =  128  =  1024  =  61440 

1  =     16  =     128  =     7680 

1  =         8  =       480 

1  =         60 

in  use,  a  ton  of  round  timber  will  often  contain  50  c.  ft.     Most  timber 
is  now  sold  by  Board  Measure,  the  "  ton"  being  nearly  obsolete. 

1  From  the  Latin,  Congiarium,  a  gallon. 

b  From  Octans,  an  eighth  part. 


72  MEASURES,  WEIGHTS,  ETC.  [ART.  IV. 

It  is  sometimes  desirable  to  make  an  estimate  of  the 
weight  of  fluids.  A  pint  of  water  weighs  a  pound;*  45 
drops  make  about  a  fluidrachm ;  a  common  teacup  holds 
about  4  fluidounces;  a  common  tablespoon  about  half  a 
fluidounce ;  a  teaspoon  about  1  fluidrachm. b 

5.  DRY  MEASURE. 

The  denominations  are  Bushels,  Pecks,  Quarts,  and 
Pints. 

bu.    pic.       qt.         pt. 

1  =  4  =  32  =  64 

1=     8=  16 

1=2 

The  United  States  standard  is  the  bushel  measure,  con 
taining  543391.89  standard  grains,  (7 7. 62 74  pounds  Avoir 
dupois,)  of  distilled  water,  at  the  temperature  of  39°. 83 
Fahrenheit,  and  barometer  30  inches  at  62°  Fahrenheit. 
This  corresponds  very  nearly  with  the  Winchester  bushel, 
which  is  a  cylinder  18^  inches  in  diameter,  and  8  inches 
deep,  containing  2150.42  cubic  inches. 

6.  TROY  WEIGHT.6 

The  denominations  are  Pounds,  Ounces,  Pennyweights, 
and  Grains. 

lb.        oz.        diet.          gr. 

1  =  12  =  240  =  5760 

1  =     20  =     480 

1=      24 

The  standard  Troy  poundd  is  equivalent  to  the  weight  of 

1  A  pint  of  distilled  water  at  62°  Fahrenheit  weighs  1  lb.  The  dif 
ference  between  distilled  water  and  well-water  at  any  ordinary  tem 
perature  is  so  slight,  that  in  making  estimates,  a  pint  of  water  may 
always  be  considered  as  weighing  a  pound. 

b  United  States  Dispensatory. 

c  "  Troy"  Weight  is  said  to  signify  London  weight,  the  name  being 
derived  from  Troy  Novant,  the  ancient  name  of  London. — McCulloch. 

d  The  Troy  pound  was  declared  to  be  "  the  standard  and  Troy  pound 
of  the  Mint  of  the  United  States,  conformably  to  which  the  coinage 
thereof  shall  be  regulated,"  by  an  act  of  Congress  of  19th  May,  1828. 


§  12.]  STANDARDS   OF  THE   UNITED    STATES.  73 

22.79442  cubic  inches  of  distilled  water  at  its  maximum 
density,  the  barometer  standing  at  30  inches;  or  to  22.8157 
c.  in.  of  water  at  62°  Fahrenheit,  barometer  at  30  inches.* 
It  was  copied  by  Captain  Kater;  in  1827,  from  the  English 
Imperial  Troy  pound. 

In  APOTHECARIES'  WEIGHT,  which  is  used  in  compound- 
Ing  medicines,  the  Troy  pound  is  subdivided  into  Ounces 
(3),  Drachms  (3),  Scruples  (B),  and  Grains. 

'»•     3      3       9        *r 

I  =  12  =  96  =  288  =  5760 

1  =    8  =    24  =    480 

1  =       3  =       60 

1  =      20 

In  DIAMOND  WEIGHT,  4  quarters  =  1  grain  ;  4  grains  = 
1  carat;  7?  carats  —  1  Troy  dwt.  A  diamond  weighing  1 
carat  is  worth  about  $9  if  rough,  and  $36  if  cut.  The  value 
increases  as  the  square  of  the  weight,  unless  the  weight  ex 
ceeds  20  carats,  in  which  case  the  increase  is  not  so  rapid. 
Thus  a  cut  diamond  weighing  3  carats,  would  be  worth 
about  32  X  36  =  324  dollars.5  The  gold  carat  grain  = 
2-J-  dwt. 

At  the  UNITED  STATES  MINT,  the  Troy  ounce  is  adopted 
as  the  standard,  and  all  weights  are  expressed  in  decimal 
multiples  and  submultiples  of  the  ounce.  Thus  951b.  8oz. 
15dwt.  15gr.  of  bullion,  would  be  credited  on  the  Mint 
books  as  1148.65625oz.c 

7.   AVOIRDUPOIS  WEIGHT. 

The  denominations  are  Tons,  Hundred-weights,  Quarters, 
Pounds,  Ounces,  and  Drams. 

This  is  the  only  direct  legislation  in  regard  to  the  adoption  of  stand 
ards,  but  the  joint  resolution  of  June  14th,  1836,  indirectly  recognises  the 
weights  and  measures  used  at  the  Custom-Houses,  as  having  been 
"  adopted  as  standards." 

»  McCulloch.  b  Encyclopaedia  Americana. 

c  Professor  McCulloh,  of  the  U.  S.  Mint. 


74                             MEASURES,  WEIGHTS,    ETC.  [ART.  IV. 

T.       ciot.*-      qr.           Ib.                oz.                  dr.  gr.  Troy. 

1  =  20  =  80  =  2240  =  35840  =  573440  =  15680000 

1  =     4  =     112  =     1792  =     28672  =  784000 

1  =      28  =       448  =       7168  =  196000 

1  =         16  =        256  =  7000 

1  =          16  =  437£ 


The  standard  Avoirdupois  pound  is  equivalent  to  7000 
Troy  grains,  or  to  the  weight  of  27.7274  c.  in.  of  distilled 
water,  at  62°  Fahrenheit,  barometer  at  30  inches.b 

In  many  of  the  states,  statutes  have  been  enacted,  fixing 
the  ton  at  2000  Ib.,  the  hundred  at  100  Ib.,  and  the  quarter 
at  25  Ib.  But  in  the  standard  of  the  general  governmeDt, 
the  ton  of  2240  Ib.  with  its  subdivisions,  is  still  retained.0 
Even  where  the  legal  ton  is  2000  Ib.,  2240  Ib.  are  often  al 
lowed  in  weighing  bulky  or  cheap  materials,  such  as  iron, 
coal,  plaster,  &c. 

8.   FEDERAL  MONEY.* 

The  denominations  are  Eagles,  Dollars,  Dimes,  Cents, 
and  Mills. 

E.        J$           di.           ct.  mi. 

1  =  10  =  100  =  1000  =  10000 

1  =     10  =     100  =  1000 

1  =       10  =  100 

1=  10 

The  only  coins  in  circulation  are  the  double  eagle,  the 


a  From  c.  for  centum  (which  signifies  one  hundred),  and  wt.  weight. 

b  The  old  English  pound,  which  is  said  to  have  been  the  legal  stan 
dard  of  weight  from  the  time  of  William  the  Conqueror,  to  that  of 
Henry  VII.,  was  derived  from  the  weight  of  grains  of  wheat ;  32  grains 
gathered  from  the  middle  of  the  ear,  and  well  dried,  made  a  peiray- 
weight,  20  pennyweights  an  ounce,  and  12  ounces  a  pound.  Henry 
VII.  altered  this  weight  and  introduced  the  present  Troy  pound,  which 
is  $  of  an  ounce  heavier  than  the  Saxon  pound.  The  Avoirdupois 
pound  was  introduced  by  a  statute  of  24  Henry  VIII. — JBrande. 

c  See  the  different  Tariff  Acts. 

d  Manual  of  Coins. 


§12.]  STANDARDS   OF  THE   UNITED   STATES.  75 

eagle,  half-eagle,  quarter-eagle,  and  dollar,  of  gold;  the 
dollar,  half-dollar,  quarter-dollar,  dime,  and  half-dime,  of 
silver ;  and  the  cent  and  half-cent,  of  copper.  The  gold  and 
silver  coins  contain  T9^  pure  metal,  and  y1^  alloy.  The  alloy 
of  gold  is  composed  of  about  \  silver  and  f  copper  (not  to 
exceed  ?  of  silver) ;  the  alloy  of  silver  is  pure  copper.  The 
metal  thus  alloyed  is  called  standard.  The  eagle  contains 
258  grains  of  standard  gold,  the  dollar  412 2  grains  of  stand 
ard  silver,  and  the  cent  168  grains  of  copper,  and  the  mul 
tiples  and  subdivisions  of  all  the  coins  the  same  proportion. 
Previous  to  1834,  the  eagle  contained  270  grains,  of  which 
22  £  grains  were  alloy.  By  the  act  of  Congress,  June  28th, 
1834,  it  was  provided  that  all  gold  coins  minted  anterior  to 
the  31st  of  July,  of  that  year,  should  be  receivable  in  all 
payments  at  the  rate  of  94.8  cents  per  pennyweight.  The 
old  eagle  is  therefore  worth  $10.665. 

By  the  same  act,  the  following  coins  were  rendered  cur 
rent  in  the  United  States  : — 

The  gold  coins  of  Great  Britain,  Portugal,  and  Brazil,  of  not 
less  than  22  carats  fine,  at  94_8_  cents  per  pennyweight. 

The  gold  coins  of  France,  _9  fine,  at  93  1  cents  per  penny 
weight. 

The  gold  coins  of  Spain,  Mexico,  and  Colombia,  of  the  fineness 
of  20  carats  37  grains,  at  88  9  cents  per  pennyweight. 

The  silver  dollars  of  Mexico,  Peru,  Chili,  and  Central  America, 
of  not  less  -weight  than  415  grains  each,  and  those  re-stamped  in. 
Brazil,  of  the  like  weight,  and  of  not  less  fineness  than  10  oz.  15 
dwt.  in  the  Troy  pound,  of  standard  silver,  at  $1.00  each. 

The  Five-franc  piece  of  France,  when  of  not  less  fineness  than 
10  oz.  16  dwt.  in  the  Troy  pound,  of  standard  silver,  and  weigh 
ing  not  less  than  384  grains  each,  at  the  rate  of  &3  cents. 

The  following  table  exhibits  nearly  the  value  of  the  prin 
cipal  gold  coins  of  different  countries.  But,  as  most  of  the 
coins  that  circulate  in  the  community  are  more  or  less 
worn,  the  current  value  is  generally  a  few  cents  less. 


76 


MEASURES,  WEIGHTS,    ETC. 


[ART.  IV. 


NAMES  OF  COINS. 

Weight. 

Contents 
in  pure 
Gold, 

Assay. 

New- 
value. 

United  States.  —  Eagle  coined  before 
July  31,  1834  . 
Do.  coined  after  July  31,   1834, 
shares  in  proportion     .     .     . 
Austrian  Dominions.  —  Souverain  . 
Double  Ducat 

dw.    gr. 
11     6 

1018 
3  14 
4  12 

247.5 

232 
78.6 
106  4 

car.   gr. 
22 

21  2H 

21  3* 
23  2f 

d.    c.    m. 

10665 

10 
3  387 
4  59  3 

Hungarian  do  

2    5? 

53  3 

23  3i 

2  29  7 

6    *>' 

115 

18  2 

4  95  7 

Max  d'or,  or  Maximilian    .    . 
Ducat  .     . 

4    4 
2    5J 

77 
52  8 

18  H 
23  2? 

331 
2  27  5 

Berne.  —  Ducat,  double  in  proportion 
Pistole       

123 
421 

45.9 
105  5 

23  1* 
21  24 

1  977 
4  54  2 

^Brazil.  —  Johannes,  4  in  proportion 
Dobraon 

18 
34  12 

759 

21  3} 
22 

17    6  4 
32  70  6 

Dobra  

18    6 

401  5 

22 

17  30  1 

Moidore,  4  in  proportion      .     . 
Crusado    ........ 
Brunswick.  —  Pistole,  double  in  pro'n. 
Ducat  . 

622 
16i 
42H 
2    5} 

152.2 
14.8 
105.7 
51  8 

22 
21  3| 
21  24 
23  0}- 

6  55  7 
638 
455  2 
2  23  1 

Cologne.  —  Ducat       

2    5£ 

526 

23  2 

2  26  7 

*  Colombia.  —  Doubloon       .... 
Denmark.  —  Ducat,  current     .     .     . 
Ducat,  specie     

17    9 

2 

2    5| 

360.5 
42.2 
52.6 

203 
21  0£ 
23  2 

15  535 
1  81  5 

2  26  7 

Christian  d'or 

4    7 

93  3 

21  3 

421 

East  India.  —  Rupee,  Bombay,  1818 
Rupee  of  Madras,  1818  .     .     . 
Pagoda,  Star 

711 
712 

2    4£ 

164.7 
165 
41  8 

2204 
22 
19 

796 

7  11 
1  79  8 

*  'England.  —  Guinea,  4  in  proportion 
Sovereign,                do.    ... 
Seven  Shilling  Piece      .     .     . 
*France.  —  Double  Louis,  coin.  b.  1786 
Louis,                                 do. 
Double  Louis,  coined  since  1786 
Louis,                         do. 
Double  Napoleon,  or  40  francs 
Napoleon,  or  20  francs  .     .     . 
Frankfort  on  the  Main.  —  Ducat      . 
Geneva.  —  Pistole,  old        .... 

5    Si 
5    24 
1  19 
10  11 
5    54 
920 
422 
8    7 
4    34 
2    5* 
4    7i 

118.7 
113.1 
39.6 
224.9 
112.4 
212.6 
106.3 
179 
89-7 
52.9 
92-5 

22 
22 
22 
21  2 
21  2 
21  24 
21  24 
21  2i 
21  2i 
23  24 
21  2 

5    75 

4838 
1  698 
968  8 
4  84  3 
9  16  2 
4  58  1 
7  703 
3866 
2279 
3  98  5 

Pistole,  new 

3  15J 

80 

21  34 

3  44  6 

Genoa,  —  Sequin   

2    5£ 

53-4 

23  3£ 

2  30  2 

Hamburg.  —  Ducat,  double  in  pro'n. 
Hanover.  —  George  d'or     .... 
Ducat 

2    5* 
4    64 

2    5J 

52-9 
92-6 
53-3 

23  24 
21  2* 
23  3£ 

2279 
399 

2  29  7 

Gold  Florin,  double  in  proportion 
Holland.  —  Double  Rvder 
Ryder       
Ducat        
Ten  Guilder  Piece,  5  do.  in  pr'n. 
Malta.  —  Double  Louis      .... 
Louis   

2    2 
1221 
6    9 
2    51 
4    7* 
1016 
5    8 

39 
283-2 
140-2 
52-8 
93-2 
215-3 
108 

18  34 
22 
22 
23  2i 
21  2i 
20  Of 
20  1 

.    1  694 

12  20  5 
6    43 
2275 
4    1  6 
9278 
4653 

Demi  Louis       ... 

2  16 

54-5 

20  1* 

2  34  8 

^Mexico.  —  Doubloons,  shares  in  pr'n. 
Milan.  —  Sequin 

17    9 

2    5f 

360-5 
53-2 

20  3 
23  3 

15535 
2  29  3 

§12.] 


STANDARDS   OP  THE   UNITED   STATES. 


77 


NAMES  OF  COINS. 

Weight 

Contents 
in  pure 
Gold. 

Assay. 

New- 
value. 

Doppia  or  Pistole  

dw.    gr 

4    14 

grains. 

88  4 

car.   gr. 

21  3 

d.    c.    m. 
3  80  7 

Forty  Lire  Piece,  1808    .     .     . 
Naples.—  Six  Ducat  Piece,  1783     . 
Two  do.  or  Sequin,  1762    .     . 
Three  do.  or  Oncetta,  1818     . 
Netherlands.  —  Gold  Lion  or  14  Florin 
Piece     .... 
Ten  Florin  Piece,  1820       .     . 
Parma.  —  Quadruple   Pistole,  double 
in  proportion       .     .     . 
Pistole  or  Doppia,  1787       .    . 
Do.               do.       1796       .     . 
Maria  Theresa,  1818       .     .     . 
Piedmont.  —  Pistole,  coined  since  1785 
half  in  proportion 
Sequin,  half  in  proportion 
Carlino,  coined  since  1785,  half 
in  proportion       .     . 

8    8 
5  16 
1  20i 
210J 

5    7* 

4    7t 

18    9 
414 
414 
4    34 

5  20 

2    5* 

29    6 

179.7 
121.9 
37.4 
58.1 

117.1 
93.2 

386 
97.4 
95.9 
89.7 

125.6 
52.9 

634  4 

21  24 
21  If 
20  H 
23  34 

22 
21  24 

21 
21  1 
203* 
21  24 

212* 

23  24 

21  2* 

7742 
5249 
1  61  3 
2496 

546 
4    1  6 

16627 
4  196 
4  135 
385  1 

541  2 
2279 

27  33  4 

Piece  of  20  Francs,  ealled  Ma- 
rengo     ...               . 

4    3i 

82  7 

20 

3  56  4 

Poland  —  Ducat 

2    5J 

52  9 

23  24 

2  27  9 

*Portugal.—  Dobraon    

34  12 

759 

22 

32  70  6 

Dobra 

18    6 

401  5 

22 

17  30  1 

Johannes       

18 

17    6  4 

Moidore,  half  in  proportion 
Piece  of  16  Testoons,  or  1600  rees 
Old  Crusado  of  400  rees      .     . 
New  Crusado  of  480  rees   .     . 
Milree,  coined  in  1755    .     .     . 
New  Dobra 

622 
2    6 
15 
16i 
19* 
17    6 

152.2 
49.3 
13.6 
14.8 
18.1 

21  3* 
22 
21  3| 
21  3* 
21  3t 
22 

6557 
2  12  1 
588 
637 
78 
16  25  3 

Joannese,  double  in  proportion 
Half  in  proportion       .... 
Piece  of  12  Testoons,  or  1200  rees 
Piece  of  8  Testoons,  or  800  rees 
Prussia.—  Ducat,  1748       .... 
Ducat,  1787       ..'... 
Frederick,  double,  1769      .     . 
Do.          do.       1800      .     . 
Do.       single,   1778      .     . 
Do.         do.       1800      .     . 
Rome.  —  Sequin,  coined  since  1760 
Scudo  of  the  Republic    .     .     . 
Russia.  —  Ducat,  1796  

9    64 
415 
1  164 
1    44 
2    5f 
2    5* 
814 
814 
4    7 
4    7 
2   44 
17    04 
2    6 

52.9 
52.6 
185 
184.5 
92.8 
92.2 
52.2 
367 
53.2 

21  3* 
21  3* 
21  3t 
21  34 
2324 
23  2 
21  24 
21  2 
21  24 
21  2 
23  34 
21  24 
23  24 

876  3 
437  1 
1  574 
1  12 
2279 
226  7 
7975 
7  95  1 
3  99  9 
3975 
225 
15804 
2  29 

Ducat,  1763       

2    5f 

52.6 

23  2 

2  26  7 

Gold  Ruble,  1756       .... 
Gold  Ruble,  1799       .... 
Gold  Poltin,  1777      .... 
Imperial,  1801   .     .     . 

1    04 

18* 
9 
7  174 

22.5 
17.1 

8.2 
181  9 

22 
21  3* 
22 
23  24 

967 
737 
35  5 

7  83  6 

Half  do.    1801  .     . 

3  20? 

90  9 

23  24 

3  91  3 

Do.   do.    1818  

4    34 

91  3 

22  04 

3  94  2 

Sardinia.  —  Carlino,  half  in  proportion 
Saxony.  —  Ducat,  1784       .... 
Ducat,  1797       .'.... 

10    74 
2    5* 
2    5* 

219.8 
52.6 
52.9 

21  14 
232 
2324 

947 
226  7 
2279 

78 


MEASURES,  WEIGHTS,  ETC. 


[ART.  iv. 


NAMES  OF  COINS. 

Weight. 

Contents 
in  pure 
Gold. 

Assay. 

New- 
value. 

Augustus,  1754      
Do           1784      

dw.    gr. 

4    6^ 
4    61 

grams^ 

92  2 

car.  gr. 
21   1| 

21  2i 

d.     c.    m. 

3  927 
3  97  4 

Sicily  —  Ounce    1751    ...         . 

2  201 

58  2 

20  H 

2  50  5 

Double  do    1758 

5  17 

117 

20  2 

542 

*  Spain.  —  Quadruple  Pistole,  or  Doub 
loons,  1772,  double  and  single, 
and  shares  in  proportion  .     . 
Doubloon,  1801      

17    81 
17    9 

372 
3605 

21  21 
20  3 

16    38 
15  53  5 

Pistole    1801 

4    8i 

90  1 

20  3 

3  88  4 

Coronilla,  Gold  Dollar,  or  Vin- 
tem    1801       

1    3 

228 

20  H 

98  3 

2    5 

51  9 

23  2 

2  23  6 

Switzerland.  —  Pistole  of  Helvetic  Re 
public,  1800     .     . 
Troves  —  Ducat 

42H 
2    5$ 

105.9 
52  6 

21  21 
23  2 

456 
2  26  7 

Turkey.  —  Sequin  Fonducli,  of  Con 
stantinople,  1773   .     . 
Do   1789 

2    5* 
2    5f 

43.3 
42  9 

19  H 
19  Of 

1  86  8 
1  84  8 

Half  Misseir,  1818      .... 
Sequin  Fonducli 

18i 

2    5 

12.2 
42  5 

16  01 
19  1 

52  1 
1  83  1 

Yermeebeshlek      

3    If 

70.3 

22  31 

328 

Tuscany.  —  Zechino,  or  Sequin  . 
Ruspone  of  the  k'm.  of  Etruria 
Venice.  —  Zechino  or  Sequin,  shares 
in  proportion  .... 
Wurtemburg.  —  Carolin     .... 
Ducat 

2    5f 
6  17i 

2    6 
6    31 
2    5 

53.6 
161 

53.6 
113.7 
51  9 

23  3f 
23  3£ 

23  3i 
182 
23  2 

230  9 
6  93  9 

231 

4  89  8 
2  23  7 

Zurich.  —  Ducat,  double,  and  half  in 
proportion  

2    5* 

52.6 

232 

226  7 

The  foregoing  Table  is  copied  from  the  American  Al 
manac  for  1835.  It  was  originally  compiled  from  the 
"  Manual  of  Coins/'  published  at  the  United  States  Mint, 
"Kelly's  Cambist,"  and  "Moore's  Philadelphia  Price  Cur 
rent." 

The  gold  coins  of  the  countries  to  which  the  star  is  pre 
fixed,  if  possessed  of  the  fineness  prescribed,  are  made,  by 
the  act  already  referred  to,  to  "pass  current  as  money,  and 
to  be  receivable  in  all  payments,  by  weight,  for  all  debts  and 
demands,  from  and  after  the  31st  day  of  July,  1834."  The 
other  coins  in  the  Table  are  not  made  a  legal  tender  j  but 
they  are  sold  at  a  certain  rate  per  dwt.,  according  to  the 
purity  of  the  gold. 


§12.]  STANDARDS   OF  THE   UNITED   STATES. 


79 


TABLE  OF  THE  PRINCIPAL  SILVER  COINS  OF  DIFFERENT 
COUNTRIES. 


NAMES  OF  COINS. 


Austria. — Rix  Dollar,  or  Florin,  Convention  .     .  179.6 

Copftsuck,  or  20  Creutzer  Piece   ....        59.4 

Halbe  Copftsuck,  or  10  Creutzer  Piece       .        28.8 

Baden.— Rix  Dollar 358.1 

Bavaria.—  Rix  Dollar  of  1800 345.6 

Copftsuck 59.4 

Brunswick. — Rix  Dollar,  Convention     ....  359.2 

Denmark.—  Ryksdaler 388.4     1 

Mark,  specie,  or  Half  Ryksdaler       .     .     .        64.4 

East  Indies. — Sicca  Rupee,  Calcutta     ....  175.9 

Company's  Rupee  (1835) 165. 

Bombay,  new,  or  Surat  (1818)      ....  164.7 

Fanam,  Cananore 32.9 

"         Bombay,  old 35. 

Pondicherry       .     , 22.8 

Gulden,  Dutch  East  India  Company      .     .  148.4 

England.— Crown  (old) 429.7     1 

Shilling  "         85.9 

Crown  (new) 403.6     1 

Shilling  "         80.7 

France. — Franc 69.4 

Genoa.— Scudo  of  8  Lire 457.4     1 

Hamburg.—  Rix  Dollar,  specie 397.5     1 

Double  Mark,  32  Schilling 210.3 

8  Schilling  Piece 50.1 

Hanov&r.—'Ka.  Dollar,  Constitution       ....  400.3     1 

Florin,  or  piece  of  f,  fine 200.3 

Holland.— Florin,  or  Guilder 146.8 

12  Stiver  Piece 92.4 

Florin  of  Batavia 141.6 

Lubeck.—  Rix  Dollar,  specie 391.9     1 

Mark        105.1 

Lucca.— Scudo 372.3     1 

Malta. — Ounce  of  Emman.  Pinto 337.4 

2  Tari  Piece 17.7 

Milan.— Scudo  of  6  Lire 319.6 

Lira 52.8 

Modena.— Scudo  of  1796 287.4 

Naples.—  Ducat,  new 295.4 

Piece  of  10  Carlini 295.1 

Netherlands.— Florin  of  1816 148.4 

Poland.— Florin,  or  Gulden 84. 

Portugal—  New  Crusado  (1809) 198.2 

Seis  vintems,  or  Piece  of  120  Rees    .     .     .        46.6 


Contents 
in  pure 


cts. 

48 
16 
08 
96 
93 
16 
96 
05 
17 
47 
44 
44 
09 
09 
06 
40 
15 
23 
09 
22 
19 
23 
07 
57 
13 
08 
54 
40 
25 
38 
().-> 
28 
00 
91 
05 
86 
14 
77 
80 
80 
40 

2;] 
63 
18 


Brande. 


80 


MEASURES,  WEIGHTS,  ETC. 


[ART.  iv. 


NAMES  OF  COINS. 


Testoon 

Portuguese  Colonies. — Piece  of  8  Macutes,  of  Por 
tuguese  Africa 
Prussia. — Rix  Dollar,  Convention 

Florin,  or  Piece  of  f 

Rome. — Scudo,  or  Crown 

Paolo        

Russia. — Rouble 

Rouble  of  Alexander  (1805) 

20  Copeck  Piece 

Sardinia. — Scudo,  or  Crown 

Saxony. — Rix  Dollar,  Convention 

Sicily. — Scudo 

Spain. — Dollar 

Sweden. — Rix  Dollar 

Switzerland. — Ecu  of  4  Franken 

Turkey.—  Piastre  of  1818 

Tuscany. — Lira 

Wurtemburg. — Rix  Dollar,  specie 

Copftsuck 


159.8 
359. 
198.4 
37-1.5 

37.2 
312.1 
278.1 

62.6 
324.7 
358.2 
348.2 
370.9 
388.5 
407.6 

67.7 

53.4 
359.1 

59.8 


9.  ASTRONOMICAL  MEASURES. 

The  denominations  of  CIRCULAR  MEASURE,  and  of 
TIME,  are  the  same  in  all  civilized  countries. 

Every  circle  is  divided  into  degrees  (°),  minutes  ('),  and 
seconds  (")•  Seconds  are  usually  subdivided  decimally. 
A  Sign  in  Astronomy,  is  30  degrees.  A  Quadrant  is  90 
degrees. 

Circ.        o  " 

1  =   360  =   21600  =  1296000 

1  =         60  =  3600 

1  =  60 

The  denominations  of  Time  are  Years,  Months,  Weeks, 
Days,  Hours,  Minutes,  and  Seconds. 

Yr.         Dy.  h.  min.  sec. 

1  =  365  =  8760  =  525600  =  31536000 

1  =   24  =   1440  =    86400 

1  =     60  =     3600 

1  =      60 


§  13.]  STANDARDS   OF  GREAT   BRITAIN.  81 

A  common  year  is  365  days.  A  bissextile,  or  leap  year, 
is  366  days.  A  Julian  year  is  365 £  days.  A  tropical, 
solar,  or  civil  year  is  365d.  5h.  48m.  49.7sec.a 

13.     STANDARDS  or  GREAT  BRITAIN." 
1.  MEASURES. 

The  linear,0  superficial,  and  cubic  measures  are  the  same 
as  in  the  United  States.  The  old  wine,  beer,  and  dry  mea 
sures  have  been  supplanted  by  the 

IMPERIAL  LIQUID  AND  DRY  MEASURE. 

The  denominations  are  Quarters,  Cooms,  Bushels,  Pecks, 
Gallons,  Pottles,  Quarts,  Pints,  and  Gills. 

Qr.      C.       bit.       pk.        gal.        pot.          qt.  pt.  gi 

1  =  2  =  8  =  32  =  64  =  128  =  256  =  512  =  2048 

1  =  4  =  16  =  32  =  64  =  128  =  256  =  1024 

1  =  4  =  8  =  16  =  32  =  64  =  256 

1=2=   4=   8=  16=  64 

1=   2=   4=   8=  32 

1=   2=   4=  16 

1=2=  8 

1=  4 

The  imperial  standard  gallon  contains  10  Ib.  Avoirdupois 
of  distilled  water,  temperature  62°,  barometer  30  inches. 
Its  capacity  is  therefore  277.274  c.  in.  The  imperial 
bushel  is  a  cylinder,  of  which  the  inner  diameter  is  18 £ 
inches,  and  the  depth  8J  in. 

The  following  denominations  of  WINE  MEASURE  have  been 
discarded  under  the  new  system,  viz.  :  the  Tun  =  2  Pipes, 
the  Pipe  or  Butt  =  2  Hhd.,  the  Hogshead  =  63  Gal.,  the 
Puncheon  =  2  Tierces,  and  the  Tierce  =  42  Gallons.  In 
the  old  BEER  MEASURE,  1  Tun  =  2  Butts,  1  Butt  =  2 
Hhd.,  1  Hogshead  =  1 }  Barrels,  1  Puncheon  =  2  Barrels,  1 

a   Somerville. 

b  McCulloch,  and  Man.  of  Coins. 

c   The  Irish  mile=  3038  yd. ;  the  Scotch  mile  =  1984  yd. 


82  MEASURES,  WEIGHTS,  ETC.  [ART.  IV- 

Bbl.  =  2  Kilderkins,  1  Kilderkin  =  2  Firkins,  1  Firkin  = 
8  Gal.  of  Ale,  or  9  Gal.  of  Beer. 

Coals  were  formerly  sold  by  the  Chaldron,  which  was  sub 
divided  into  Vats,  Sacks,  and  Bushels.  The  coal  bushel 
held  1  qt.  more  than  the  Winchester  bushel.  Twenty-one 
chaldrons  made  a  Score. 

Score.  Chal.      Vats.      Sacks.        Bu. 

1  =  21  =   84  =  252  =  756 

1  =     4  =     12  =  36 

1  =       3  =  9 

1  =  3 

In  DRY  MEASURE,  a  Last  is  2  Weys,  and  a  Wey  or  Load 
is  5  Quarters. 

2.  WEIGHTS. 

All  weights  are  derived  from  the  Troy  and  Avoirdupois 
pounds.  The  Imperial  standards,  and  the  denominations  of 
Troy,  Apothecaries',  and  Avoirdupois  weight,  are  the  same 
as  in  the  United  States. 

Not  only  have  the  English  no  natural  standard  of  weight, 
but  at  the  present  time  they  have  no  standard,  the  Imperial 
Troy  pound  having  been  destroyed1  by  the  fire  which  con 
sumed  the  Houses  of  Parliament,  Oct.  16,  1834.b  But  the 
bulk  of  water  to  which  it  is  equivalent  has  been  so  accu 
rately  determined  (27.7274  c.  in.),  that  it  could  easily  be 
restored.  The  length  of  the  seconds'  pendulum  at  Green 
wich,  may,  therefore,  be  very  properly  regarded  as  the  present 
basis  of  the  entire  system  of  weights,  measures,  and  cur 
rencies,  both  of  Great  Britain  and  the  United  States. 

Avoirdupois  weight  may  be  readily  converted  into  Troy, 
or  Troy  into  Avoirdupois. 

144  lb.  Avoirdupois  =  175  Ib.  Troy. 
192  oz.  Avoirdupois  =175  oz.  Troy. 

A  stone  is  generally  14  lb.  Avoirdupois.     But  a  stone  of 
*  Brande.  b  Wade. 


§  13.]  STANDARDS   OP  GREAT  BRITAIN.  83 

butcher's  meat,  or  fish,  is  8  Ib.  A  stone  of  glass  is  5  Ib. 
A  seain  of  glass  is  24  stone.  A  truss  of  hay  =  56  Ib.  A 
truss  of  new  hay,  until  the  1st  of  September  =  60  Ib.  A 
truss  of  straw  =  36  Ib.  36  trusses  make  a  load. 

In  weighing  wool,  1  last  =  12  sacks,  1  sack  =  2  weys,  1 
wey  —  62-  tods,  1  tod  =  2  stone,  1  stone  =  2  cloves,  1  clove 
=  7  Ib.  A  pack  of  wool  contains  240  Ib. 

8  Ib.  =  1  clove  of  cheese  or  butter,  and  56  Ib.  =  1  fir 
kin  of  butter.  A  wey  is  32  cloves  in  Essex,  or  42  cloves 
in  Suffolk. 

3.  ENGLISH  OR  STERLING  MONEY. 

The  denominations  are  Pounds,  Shillings,  Pence,  and 
Quarters,  or  Farthings. 

£  s.  d.  qr. 

1  =  20  =  240  =   960 

1  =     12  =     48 

1  =       4 

A  guinea  is  21  shillings.  A  crown  is  5  shillings.  The 
coin  which  represents  the  pound  is  called  a  sovereign.  The 
coins  are  the  five-guinea  piece,  the  guinea,  half-guinea,  quar 
ter-guinea,  seven-shilling  piece,  double  sovereign,  sovereign, 
and  half-sovereign  of  gold, — the  crown,  half-crown,  shil 
ling,  sixpence,  fourpence,  threepence,  twopence,  one-and-a- 
halfpence,  and  penny  of  silver, — and  the  penny,  halfpenny, 
farthing,  and  half-farthing  of  copper.a 

A  pound  of  silver  of  the  English  mint  standard,  contains 
lloz.  2dwt.  of  pure  silver,  and  18dwt.  of  alloy.  This 
pound  is  coined  into  66  shillings.  A  shilling  therefore 
weighs  87.27  grains,  and  contains  80.727  grains  pure  silver. 
The  standard  for  gold  is  11  parts  fine  gold  and  1  part  alloy. 
The  sovereign  weighs  123.274  grains,  and  contains  113.001 

a  In  Scotland,  a  bodle=  ?  of  an  achison=  I  of  a  lawlee=  I  of  a 
plack  =i  £  of  a  penny. — Gregory. 


84  MEASURES,  WEIGHTS,  ETC.  [ART.  IV. 

grains  of  pure  gold.  A  sovereign  of  full  weight  is  there 
fore  worth  $4.866.a  The  guinea  and  its  subdivisions  have 
not  been  coined  since  1816. 


14. — STANDARDS  OF  FRANCE.* 
1.  MEASURES. 

The  standard  unit  of  linear  measure  is  the  Metre,  from 
which  all  measures  and  weights  are  derived.  It  is  intended 
to  be  equivalent  to  y^^o^u  °^  ^e  distance  from  the  pole 
to  the  equator,  and  was  determined  by  measuring  the  dis 
tance  from  Dunkirk  to  Rhodes.0  The  names  of  the  multi 
ples  and  submultiples  of  the  unit,  in  all  the  tables,  are 
formed  by  the  following  prefixes  : 

Deca  prefixed,  signifies         10  times.     Deci  denotes   -j^. 

Hecto       "  "  100     "  Centi     "      TJ<j. 

Kilo        "  "  1000     «         MiUi     "    TTtaj. 

Myria     "  "          10000     " 

Thus  the  Decametre  =  10   Metres;    the  Decistere  =  -J^ 
Stere ;  the  Hectogramme  =•  100  Grammes. 

The  metre  is  equivalent  to  39.371  inches.  The  milli 
metre  is  sometimes  called  trait  (line),  the  centimetre,  doigt 
(finger),  the  decimetre,  palme,  and  the  decametre,  perche. 

The  unit  of  superficial  measure  is  the  Are,  which  is  a 
square  decametre,  and  is  equivalent  to  119.6046  square 
yards.  The  hectare  is  often  called  arpent  (acre). 

a  In  1838  a  dispute  arose  in  the  settlement  of  an  account,  which  was 
submitted  to  arbitration.  The  charges  were  all  made  in  English 
money,  and  the  point  in  dispute  was  the  value  of  the  pound  sterling, 
in  our  own  currency.  It  was  decided  by  the  referees,  to  be  $4.858. — 
Mer.  Mag. 

At  the  Custom-House,  the  sovereign  is  estimated  at  $4.84;  the  pound 
of  the  British  Provinces,  Nova  Scotia,  New  Brunswick,  Newfound 
land  and  Canada,  at  $4.00;  the  pound  of  Jamaica,  Honduras,  and 
Turk's  Island,  at  $3.00  ;  the  pound  of  Nassau,  at  $2.50. 

b  McCulloch,  Enc.  Amer.,  Hunt's  Mer.  Mag. 

•  About  570  miles. — Hunt's  Mer.  Mag. 


§  14.]  STANDARDS   OF  FRANCE.  85 

The  unit  of  solid  measure  is  the  Stere,  which  is  a  cubic 
metre,  and  is  equivalent  to  35.3174  cubic  feet,  or  1.30805 
cubic  yards. 

The  unit  of  measures  of  capacity  is  the  Litre,  which  is  a 
cubic  decimetre,  and  is  equivalent  to  1.05676  quarts  of  our 
standard.  The  setter  is  equivalent  to  the  hectolitre,  and  the 
muidj  or  barrel,  to  a  kilolitre. 

In  the  Systeme  Usuel,a  the  names  of  the  ancient  weights 
and  measures  are  retained,  together  with  the  subdivisions 
into  halves,  quarters,  eighths,  &c.  The  toise  usuelle  =  2 
metres;  the  pied  or  foot  =  £  metre;  the  aune  =  1-J  me 
tres;  the  boisseau  =  12.5  litres. 

2.  WEIGHTS. 

The  unit  of  weight  is  the  Gramme,  which  is  derived  from 
the  cubic  centimetre,  and  is  equivalent  to  15.434  Troy 
grains.  In  the  Systeme  Usuel,  the  half-kilogramme  is 
called  a  livre,  which  is  thus  subdivided  :  4  gros  make  1 
once ;  16  onces  make  1  livre  usuelle ;  2  livres  make  1  kilo 
gramme.  A  millier  =  1000  kilogrammes  =  2205.48  Ib. 
It  is  used  for  marine  tonnage. 

3.  MONEY. 

Accounts  are  kept  in  Francs  and  Centimes.  The  old 
denomination,  Sols  or  Sous,  is  sometimes  used  in  accounts, 
20  sous  being  rated  as  a  franc.  Ten  centimes  make  a 
decime. 

F.       D.        c. 
1  =  10  =  100 
1  =     10 

The  subdivision  of  the  franc,  therefore,  resembles  our 
subdivision  of  the  dollar  into  dimes  and  cents.  Prior  tc 
the  revolution,  the  money  was  divided  into  louis-d'ors, 


a  Called  "  usuel,"  because  the  common  subdivisions  are  retained 
but  their  value  is  altered  to  correspond  with  the  new  standard. 


86  MEASURES,  WEIGHTS,  ETC.  [ART.  IV. 

ecus  or  crowns,  livres  tournois,  sous,  and  deniers ;  the  livre, 
when  newly  coined,  being  equivalent  to  the  franc. 

L.d'or.    e.        liv.  s.  d. 

I  =  4  =  24  =  480  =  5760 

1  =     6  =  120  =  1440 

1  =    20  =    240 

1  =      12 

The  gold  coins  now  in  circulation,  are  the  double  louis- 
d'or,  the  louis-d'or,  the  double  Napoleon  (worth  40  francs), 
the  Napoleon,  the  40  franc  piece,  and  the  20  franc  piece ; 
the  silver  coins  are  the  crown,  half-crown,  30  sous,  15  sous, 
6  livres,  5  francs,  2  francs,  franc,  and  quarter-franc  j  the  cop 
per  coins  are  the  5  sous,  2  sous,  sol,  decime,  5  centimes,  2  cen 
times,  and  centime.  The  mint  standard  for  both  gold  and 
silver,  is  T9<j  fine  metal,  and  -j1^  alloy.  A  kilogramme  of 
standard  gold  is  coined  into  155  twenty-franc  pieces.  A 
kilogramme  of  standard  silver  is  coined  into  200  francs. 
The  Custom-House  valuation  of  the  franc  is  $0.186. 

4.  SUBDIVISIONS  OF  THE  CIRCLE  AND  OF  TIME. 

With  the  introduction  of  the  decimal  system  of  measures, 
weights,  and  money,  an  attempt  was  also  made  to  change 
the  divisions  of  the  circle  and  of  time.  Each  quadrant  of 
the  circle  was  divided  into  100  degrees  (making  400°  in 
the  entire  circle,  instead  of  360°),  each  degree  into  100 
minutes,  and  each  minute  into  100  seconds.  A  series  of 
logarithmic  trigonometrical  tables,  to  correspond  with  this 
system,  was  computed  by  M.  Borda,  and  a  compendium  of 
his  work,  with  the  logarithms  extended  to  seven  places,  has 
been  published. 

Each  of  the  12  months  was  composed  of  three  decades  of 
10  days  each,  and  at  the  end  of  the  year,  five  intercalary 
days,  (in  leap  years  six  days,)  were  added.  The  day  was 
divided  into  10  hours,  the  hour  into  100  minutes,  and  the 
minute  into  100  seconds.  After  a  short  trial,  the  attempt 
to  introduce  these  changes  entirely  failed. 


§15.]  MISCELLANEOUS   TABLE.  87 

1 5.    MISCELLANEOUS  TABLE  a  or  MEASURES,  WEIGHTS, 
AND  MONEYS  or  ACCOUNT. 

1.  ACAPULCO. — See  MEXICO. 

2.  ALEXANDRIA. — The  yard  or pik  =  26.8  inches.    The  measures 
for  corn  are  the  rhebebe  =  4.364  Eng.  bushels,  and  the  quillot  or 
kisloz  —  4. 729  bushels.    The  cantaro  or  quintal  =100  rottoli  ;  but  the 
rottolo  has  four  different  values.     The  rottolo  forforo  =  .93471b. 
Av. ;  1  rottolo  zaidino  =  1.335  Ib.  av. ;  1  rottolo  zauro  =  2.07  Ib.  Av. ; 
1  rottolo  mina  =  1.67  Ib.  Av. 

MONEY. — Accounts  are  kept  in  current  piastres.  1  piastre  =  40 
paras  or  medini;  1  medino  =  30  aspers,  or  8  borbi,  or  6  forli.  A 
purse  contains  25000  medini.  A  piastre  is  worth  about  6  cents. 
Large  payments  are  generally  made  in  Spanish  dollars. 

3.  ALICANT. — The  yard  or  vara  =  4  palmos  —  29. 96  inches.     In 
liquid  measure  1  cantaro  =  8  medios  =  16  quartillos  —  3.05  Eng.  wine 
gallons.    The  tonnelada  or  ton  =  2  pipes  =  80  arrobas  =  100  canta- 
ros.     In  dry  measure  1  cahiz  =  12  barchillas  =  96  medios  =  192 
quartillos  =  7  Winch,  bush.    The  cargo  =  2J  quintals  =  10  arrobas. 
The  arroba  =  27  Ib.  6oz.  Av.,  and  contains  24  large  pounds  of  18 
Castilian  ounces,  or  36  small  pounds  of  12oz.  each.     At  the  Cus- 
tom-House,  the  arroba  =  25  Ib.  of  16oz.  each. 

MONEY. — Accounts  are  kept  in  libras  of  20  sueldos,  each  sueldo 
containing  12  dineros.  The  libra  or  peso  =  10  reals  =  272  maravedis 
of  plate  or  512  maravedis  vellon  =  78  cents. 

4.  AMSTERDAM. — In  1820,  the  French  system  of  measures  and 
weights  was  introduced  into  the  Netherlands,  the  names  only  being 
changed.     The  unit  of  LONG  MEASURE  is  the  elle,  which  equals  the 
French   metre.     Its    decimal    divisions    are  the  palm,  duim,  and 
streep   (corresponding   to    the   decimetre,  centimetre,  and   milli 
metre),  and  its  decimal  multiples,  the  roede  and  mijle  (correspond 
ing  to  the  decametre  and  kilometre).     The  unit  of  SQUARE  MEA 
SURE  is  the  vierkante  elle  or  square  elle,  which  equals  the  French 
centiare  or  metre  carre.     Its  divisions  and  multiples  are  the  vier 
kante  palm,  vierkante  duim,  vierkante  streep,  vierkante  roede,&nd  vier 
kante  bunder.     The  vierkante  bunder  =  1  Are.     In  MEASURES  OP 
CAPACITY  1  kubicke  elle  =  1  stere.     Its  divisions  are  the  kubicke 
palm,  duim,  and  streep.     A  k.  elle  of  firewood  is  called  wisse.     In 

1  McCulloch,  Hunt's  Mer.  Mag.,  Am.  AL,  Boston  Custom-House 
Table,  Enc.  Amer. 


88  MEASURES^  WEIGHTS,  ETC.  [ART.  IV. 

DRY  MEASURE,  1  hop  =  1  litre.  The  maatje,  schepel,  and  mudde,  or 
zak,  correspond  to  the  decilitre,  decalitre,  and  hectolitre.  In  LIQUID 
MEASURE,  the  kan,  maatje,  vingcrhocde,  and  vat,  are  respectively 
equivalent  to  the  litre,  decilitre,  centilitre,  and  hectolitre.  The  last, 
or  measure  for  corn,  =  27  muddcn.  The  aam,  liquid  measure,  =  4 
ankers  =  8  steckans  =  2l  viertels  =  64  stoopen  =  128  mingles  =  256 
/unto  =  180  litres. 

The  WEIGHTS  are  the  wigtje,  korrel,  lood,  ons,  and  pond,  corre 
sponding  to  the  gramme,  decigramme,  decagramme,  hectogramme,  and 
kilogramme.  The  last  for  marine  tonnage  =  2000  ponds.  The 
apothecary's  new  pound  =  5787  grains  Troy,  or  375  grammes,  and 
is  subdivided  as  the  English  apothecary's  pound,  into  ounces, 
drams,  scruples,  and  grains.  By  the  old  method  of  calculating, 
100  Ib.  Amsterdam  =  108.923  Ib.  Avoirdupois. 

MONEY. — 100  cents  =  1  florin  =  §0.40.  Accounts  are  sometimes 
kept  in  Flemish  money.  1  pound  =  G  florins  =  20  schillings  =  120 
stivers  =  240  groats  =  1920  pennings. 

5.  ANTWERP. — The  same  as  Amsterdam.     Of  the  old  weights, 
which  are  still  occasionally  referred  to,  the  quintal  of  100  Ib.  = 
103^  Ib.  Av.      A  schippound  is  3  quintals.     A  stone  is  8  pounds. 
Of  the  old  measures,  1  last  =  372-  viertels  =  150  macken  =  about  9|- 
imperial  quarters.     A  barrel  =  26J  Eng.  gallons. 

6.  ARABIA. — See  BUSSORAH,  DJIDDA,  MOCHA,  MUSCAT. 

7.  AUSTRIA. — See  TRIESTE. 

8.  BANGKOK. — 1  fathom  =  4  cubits  =  8  spans  =  96  finger-breadths 
=  about  6J  ft.  Eng.     20  fathoms  =  1  sen,  and  100  sen  =  1  yuta. 
In  weighing,  1  picul  =  50  catties  =  1334-  Ib.  av. 

MONEY. — The  currency  consists  only  of  silver  and  cowrie  shells. 
1  bat  or  tical  —  4  salungs  =  Sfuanas  =  16  sing-p'hais  =  32  p'hai- 
nungs  —  fAQO  bia  or  cowries—  55  cents  nearly.  80  ticals  make  1 
catty,  and  100  catties  make  1  picul.  Gold  and  silver  are  weighed 
by  small  weights  which  have  the  same  denominations  as  the  coins. 
The  p'hai-nung  is  then  divided  into  32  sagas. 

9.  BARCELONA. — The  yard  or  cana  =  8  palmos  =  32  quartos  =  21 
inches  nearly.     The  quartera,  or  measure  for  grain  =  12  cortanes  = 
4Spicolins  =  .235  Winch,  quarters.     The  carga  or  liquid  measure  = 
12  cortanes  or  arrobas  =  24:  cortarinas  =  72  mitadellas  =  32.7  wine 
gallons.     4  cargas  =  1  pipe. 

MONEY. — 1  libra  =  20  sueldos  —  240  dineros  =  480  mallas  =  53 


I     .       . 
§15.]  MISCELLANEOUS   TAD^E.  89 

cents.    The  libra  is  likewise  divided  into  6f  reales  deplata  Catalan, 
or  10  reales  ardites. 

10.  BATAVIA. — The   Chinese  weights   are  used,  (the  picul  and 
catty),  but  the  picul  is  considered  equal  to  136  Ib.  Av.    Accounts 
are  kept  in  florins  or  guilders,  and  centimes.     The  rupee  =$0.44. 
The    rix   dollar  =  48  stivers  =  $0.75.       See    AMSTERDAM    and 
CANTON. 

11.  BELGIUM. — See  ANTWERP. 

12.  BENGAL.     See  CALCUTTA  and  MADRAS. 

13.  BILBAO. — See  CADIZ. 

14.  BOMBAY. — 1  guz  =  1£  haths  =  24  tussoos  =  27  inches.     In 
salt  measure,  1  rash  =  16  annas  =  1600  parahs  =  16800  adowlies 
=  2572176  cub.  inches  or  40  tons.     In  grain  measure,  1  candy  =  8 
parahs  =  56    pailies  =  224   seers  =  448    tipprees  =  1561b.    12oz. 
12.8dr.  av.     In  liquor  measure,  I  maund  =  50  seers  —  3000  rupees 
=  761b.  lloz.  13dr.  Av.    In  weighing  all  heavy  goods  except  salt,  1 
maund  =  40  seers  =  2880  tanks  =  281b.  Av.     In  pearl  weight,  1 
tank  =  24  ruttees  =  330  tuckas  =  72  Troy  grains.    In  gold  and  sil 
ver  weight,  1  tola  =  40  walls  =  179  gr.  Troy. 

MONEY. — Accounts  are  kept  in  rupees.  1  rupee  =  4  quarters  = 
400  reas  =16  annas  =  50  pice  —  §0.45.  An  urdee  is  2  reas ;  a 
doreea,  6  reas;  a  dooganey,  4  reas;  nftiddea,  8  reas;  &paunchea,  5 
rupees;  a  gold  mohur,  15  rupees.  The  annas  and  reas  are  imagi 
nary  moneys. 

15.  BRAZIL. — Same  as  LISBON. 

16.  BREMEN.— 1  ell  =  2  feet  =  22.76  Eng.  inches.     1  last  = 
4  quarts  =  40  sche/els  =  160  viertels  =  Q4Q8pints  =  80.7  Winches 
ter  bushels.     1  oxhoft  —  1J  tierces  =  6  ankers  =  30  viertels  =  264 
quarts  =  58  Eng.  wine  gal.     An  ohm  =  4  ankers.     The  commer 
cial  pound  =  2  marks  =  16  ounces  =  32  loths.     100  Ib.  Bremen 
=  109.8  Ib.  Av.     A  shippound  =  2£  centners  =  290  Ib.     A  waage 
of  iron  =  120  Ib. 

MONEY. — 1  thaler  or  rix  dollar  =  72  grootes  =  360  swarcs  = 
$0.78f. 

17.  BUENOS  AYRES. — The  same  as  CADIZ. 

18.  BURMAH. — See  RANGOON. 

19.  BUSIIIRE. — The  league  or  parasang  =  3m.  3fur.  25r.     The 
royal  guz,  or  cubit  =  37J  inches.     The  common  guz  =  25  inches. 


90  MEASURES,  WEIGHTS,  ETC.  [ART.  IV. 

The  artaba  or  principal  corn  measure  =  16  bushels  nearly. 
Pearls  are  weighed  by  the  abbas  =  2 Jgr.  Troy ;  gold  and  silver 
by  the  miscal  =  3dwt.  very  nearly.  The  maund  shaiv  —  2  maunds 
tabree  =  l3%lb.  Av.  at  the  Custom-House,  or  12£ Ib.  at  the  bazaar. 
It  is  used  by  dealers  in  sugar,  coffee,  copper,  and  all  sorts  of 
drugs.  The  maund  copra  is  7f  Ib.  at  the  Custom-House,  and  from 
7£  to  7  £  Ib.  at  the  bazaar.  It  is  used  by  dealers  in  rice  and  other 
provisions. 

MONEY. — 1  toman  =  5Q  classes  =  100  mamoodis  =  about  $2.50. 
The  toman  of  Gombroon  =$5. 

20.  BUSSORAH. — The   Arabian  mile  =  2148yd.      The  Aleppo 
yard,  for  silks  and  woollens  =  2ft.  2.4in.  ;  the  Hadded  yard,  for 
cottons  and  linens,  =  2ft.  10. 2in. ;  the  Bagdad  yard,  for  all  pur 
poses,  =  2ft.  7. Gin.    Gold  and  silver  are  weighed  by  the  cheki— 100 
miscals  =  7200gr.  Troy.     1  oke  of  Bagdad  —  2$  vakias  =  47£oz. 
Av. ;  1  maund  atteree  =  28%  Ib.  Av. ;  1  maund  sofy  or  sesse  —  90  J  Ib. 
Av. ;  1  cutra  of  indigo  =  138  Ib.  15oz.  Av.     These  are  the  weights 
used  by  the  European  merchants  settled  at  Bussorah  ;  they  differ 
a  little  from  those  used  by  the  Arabians. 

MONEY. — Accounts  are  kept  in  mamoodis  of  10  danims,  or  100 
floose.  1  toman  —  100  mamoodis  =  §8  nearly. 

21.  CADIZ.— 100  yards  or  varas  =  92J  English  yards.     The 
common  legua  =  800  varas  ;  the  legal  legua  =  500  varas.     In  corn 
measure  1  cahiz  =  12  fanegas  =  144  celeminas  =  576  quartillas  = 
1.576  bushels.    In  liquid  measure  1  cantaro  or  arroba  =  8  azumbres 
=  32  quartillos.    There  are  two  arrobas,  the  greater  and  the  less, 
the  former  =  4£,  the  latter  =  3|  wine  gallons.     A  moyo  of  wine 
=  16  arrobas.     A  botta  =  SO  arrobas  of  wine,  or  38 \  of  oil.     A 
pipe  =  27  arrobas  of  wine,  or  34£  of  oil.    100  Ib.  Castile  =  101  ^lb. 
Av.     The  ordinary  quintal  is  divided  into  4  arrobas,  or  100  Ibs. 
of  2  marcs  each. 

MONEY. — Accounts  are  kept  by  the  real  of  old  plate,  of  which 
there  are  lOf  in  the  peso  duro  or  hard  dollar.  The  real  =  16  quintos 
or  34  maravedis.  The  ducado  de  plata,  or  ducat  of  plate,  is  worth 
11  reals.  At  the  U.  S.  Custom-House  the  real  of  plate  is  esti 
mated  at  $0.10,  and  the  real  vellon  at  $0.05. 

22.  CAGLIAKI. — The palm=  10 J  inches.     The  starello,  or  corn 
measure  =  Ibu.  l£pk.  Eng.    1  cantaro  =  4  rubbi=>.  104 Ibs.  =  1248oz. 
«=  93  Ib.  Ooz.  8dr.  Av. 


§  15.]  MISCELLANEOUS   TABLE.  91 

MONEY.— 1  lira=  4  reali='2Q  soldi=  $0.186.  19  reali  make 
1  scudo. 

23.  CALCUTTA.— 1  coss=  WOO  fathoms  =  4000  cubits  =  SOW  spans 
=  24000  hands  =  96000  fingers  =  288000  barleycorns  or  jows—  1m. 
Ifur.  3T7yyd.  In  CLOTH  MEASURE  1  guz  =  2  hauts  or  cubits  =  16 
ffheriahs  =  48  angullas  =  144  jorbes  =  lyd.  Eng.  In  SQUARE  MEAS 
URE  I  biggah  =  2Q  co«afo  =  320  c/u'ttac£s=1440  sq.  ft.  A  chit- 
tack  is  5  cubits  or  hauts  in  length,  and  4  in  breadth.  In  GRAIN 
MEASURE  1  khahoon  =  16  soallies  =  3200  pallies  =  12800  raiks  = 
61200  khaonks  =  3Q  bazaar  maunds.  In  LIQUID  MEASURE  1  bazaar 
maund=  8  pussarees  or  measures  =  40  seers  =  160 pouahs  or  /»zce  =  640 
chittacks  =  3200  sicca  weight.  WEIGHTS. — 1  maund  =  40  seers  =  640 
chittacks  =  3200  st'ccas.  The  factory  maund  =  74  Ib.  lOoz.  lOfdr. 
Av. ;  the  bazaar  maund  =  82  Ib.  2oz.  l-^5  dr.  In  GOLD  AND  SILVER 
WEIGHT  1  anna  =  6^  rutties=2o  dhans  or graim=  IQQpvnkhos ;  1  sicca, 
=  10  massas  =  8Q  rutties ;  1  tolah=100  rwttz'es  =  224.588gr.  Troy. 
1  mohur  =  lGG%  rutties. 

MONEY. — Accounts  are  kept  in  sicca,  or  in  current  rupees,  with 
their  subdivisions,  annas  and  pice.  The  sicca  rupees  bear  a  batta 
(premium)  of  16  per  cent,  over  the  current.  1  gold  mohur—\G 
sicca  rupees  =  64  cahauns  =  256  annas  =  3072  pice  or  1024  punns  = 
20480  gundas.  4  cowries  (a  species  of  shell)  make  1  gunda,  and 
2560  cowries  =  1  current  rupee.  A  current  rupee  is  worth  about 
44  cents,  and  a  sicca  rupee,  $0.50.  A  lac  of  rupees  =  100000 
rupees.  A  crore  =  100  lacs. 

24.  CANADA. — See  QUEBEC. 

25.  CANTON.— 1   li=  180  fathoms  =  1800  Chinese  /ce<  =  1897£ 
Eng.  ft.     1  covid  or  cobre  — 10  punts  =  14f  inches.     There  are  no 
liquid  or  dry  measures  ;   all  articles  that  are  usually  sold  by  those 
measures,  being  sold  in  Canton  by  weight.     1  picul  —  100  catties 
or  gins  =1600  taels  or  lyangs  —  16000  mace  or  tchens  — 160000  can- 
darines  or  fivans  =  1600000  cash  or  Us  =  133i  Ib.  Av.     The  mace, 
candarine,  and  cash  are  money  weights. 

MONEY. — Accounts  are  kept  in  taels,  mace,  candarines,  and 
cash.  The  cash  is  the  only  coin  made  in  China.  It  is  composed 
of  6  parts  of  copper  and  4  of  lead,  and  is  cast  with  a  square  hole  in 
the  middle,  so  as  to  be  strung  on  a  wire  or  string.  The  circula 
ting  medium  consists  principally  of  cut  Spanish  dollars.  In  cal 
culations  of  prices,  and  of  accounts  between  foreigners  and  native 
merchants,  720  taels  =  $1000.  In  weighing  money  for  payments, 
715  taels  =  $1000,  except  to  the  Company's  treasury,  when  718 


92  MEASURES,  WEIGHTS,  ETC.  [ART.  IV 

taels  =  $1000,  or  to  native  merchants,  not  of  the  co-hong,  or  to 
ship  and  house  compradors,  when  715  taels  =  §1000.  A  tael  of  fine 
silver  should  be  worth  1000  cash,  but  on  account  of  their  conve 
nience  their  price  is  often  so  much  raised  that  only  750  are  given 
for  a  tael.  In  the  Custom-House  estimate,  the  tael  =  $1.48. 

26.  CAPE  TOWN.— 12  Rhynland  inches  =  1  Rhynland  foot ;    27 
Rhynland  inches  =  1  Dutch  ell.    In  square  measure,  1  morgen  =  600 
roods  =  86-400  square  feet  =  12441600  square  inches.    In  corn  meas 
ure,  1  load  =  10  muids  =  40  schepels  =  30bu.  2pk.  5qt.  very  nearly. 
In  liquid  measure  1  leaguer  =  4  cams  =  16  ankers  =  256  flasks  = 
152  wine  gallons.     The  weights  are  derived  from  the  standard 
pound  of  Amsterdam,  the  loot,  =  ^   °f  a  Dutch  pound,   being 
regarded  as  the  unit. 

MONEY. — Accounts  are  kept  either  in  pounds,  shillings,  pence, 
and  farthings,  (as  in  Great  Britain,)  or  in  rix  dollars,  schillings, 
and  stivers.  1  rix  dollar  =  8  schillings  =  48  stivers  =  Is.  6d. 

27.  CEYLON. — See  COLUMBO. 

28.  CHILI. — See  VALPARAISO. 

29.  CHINA. — See  CANTON. 

30.  CHRISTIANIA. — Measures  and  weights,  same  as  at  Copen 
hagen. 

MONEY. — 1  species  dollar  =  120  shillings  =  $1.06.  There  are  no 
gold  coins  made  in  Norway. 

31.  CIVITA  VECCHIA. — The  Roman  foot  =  11.72  inches  English. 
The    canna  =  78.34in.      The   builders'    canna  =  87.96in.      The 
barrel  =  12.841  imp.   gallons  of  wine,  or  12.64  imp.  gals,  of  oil. 
The  soma  of  oil  =  36. 13  imp.  gals.     The  rubbio  of  corn  =-8. 143 
imp.  gals.     The  libra  or  pound  =  12  onci=  6912  grani  =  5234gr. 
Troy.     There  are  three  cantaros  or  quintals, — of  100,  160,  and 
250  Ib.     The  migliajo  =  1000  libre. 

MONEY. — 1  scudo  =  Wpaoli=10Q  bajocchi  =  $I.QO. 

32.  COLUMBO. — The  principal  dry  measures  are  the  seer,  which 
is  a  perfect  cylinder,  4.35in.  deep,  and  4.35in.   diameter, — and 
the  parrah,  which  is  a  perfect  cube,  its  internal  dimensions  being 
11.57in.  on  every  side.     The  liquid   measure,  the  weights,  and 
the  money,  are  the  same  as  in  Great  Britain.     A  leaguer  or  legger 
=  150  gallons.     A  candy  or  bahar  =  5QQlb.  Av.     A  rix  dollar  = 
Is.  6d. 

33.  CONSTANTINOPLE. — Thepik  or  pike  is  generally  estimated  at 


§15.]  MISCELLANEOUS   TABLE.  93 

f  of  a  yard  Eug.  The  berri=  1 826yd. ;  the  Turkish  mile  =  1409yd. 
In  corn  measure  1  fortin  =  4  &M/02=  3.764  bushels.  Oil  and  other 
liquids  are  sold  by  the  alma  or  meter  —  Igal.  3pt. 

WEIGHTS. — 1  rottolo  =176  drams;  1  oke  =  2.272  rottoli;  1  quintal 
or  cantaro  =  7£  batmans  =  44  okes  =  124  Ib.  7?oz.  Av.  The  quintal 
of  cotton  is  45  okes  =  127. 2  Ib.  Av. 

MONEY. — 1  piastre  =40  paras  =  120  aspers  =  about  4  cents. 
The  piastre  is  exceedingly  variable  in  its  value,  those  coined  in 
1764  being  worth  §0.60,  and  those  coined  in  1832  being  worth 
only  §0.03.  A  bag  of  silver  =  500  piastres.  A  bag  of  gold  = 
30000  piastres. 

34.  COPENHAGEN.— The  Danish  ell  =  2  Rhindand  feet  =  about 
25  inches.     The  Danish  mile  =  8244yd.     In  dry  measure  1  last 
=  12  tocndes  or  tons  =  96  scheffels  =  384  viertels  =  47 £  bushels. 
In  liquid  measure  1  quarter  =  2  pipes  =  4  hogsheads  =  6  ahms  or 
ohms  =  24  ankers  =  240  gallons,  very  nearly.     A  fuder  of  wine 
=  930  pots  =  237£  gallons.     1  shippound  =  20   lispounds  =  320 
pounds.  =  352.8  Ib.  Av. 

MONEY. — 1  rix  dollar  =  6  marcs  =  96  shillings.  The  rigsbank 
dollar  =  §0.52.  But  the  money  generally  used  in  commercial 
transactions  is  bank  money,  which  is  at  a  heavy  discount.  The 
old  rix  dollar,  or  species  daler,  is  worth  §1.05. 

35.  CUBA.— See  HAVANA. 

36.  DANTZIC.— 1  ell  =  2  Dantzicfeet  =  22.6  Eng.  inches.    The 
Rhindand  or  Prussian  foot  =  12.356  Eng.  in.     The  Prussian  or 
Berlin    ell  =  25  J    Prussian   inches.      The  Prussian    mile  =  4.8 
English  miles.    The  last  of  grain  =  3|  mailers  =  60  scheffels  =  240 
viertels  =  960  melzen  =  91  bushels.     The  last  of  beer  =  2  fuder  = 
4  both  =  8  hogsheads  =  12  aAms  =  48  flwfo™  =  240  ?uarte  =  620gals. 
1  pipe  =  2  ahms.     The  ahm  of  wine  =  39f  gallons.     I  lispound  = 
16 J  pounds  =  264  ounces  =  8448  /o^As  =  17  Ib.  Av.     100  Ib.  Dantzio 
=  103. 3  Ib.  Av.     1  shippound  =  3  centners  =  330  pounds. 

MONEY.— 1  Mafer  or  dollar  =  30  sz7t<er  groschen  =  360  pfennings 
—  §0.69.  Accounts  are  still  sometimes  kept  in  guldens,  guilders, 
or  florins.  1  riz  rfotf arr  =  3  florins  =  90  groschen  =  270  schillings  =' 
1620  pfennings  =  §0.49. 

37.  DENMARK. — See  COPENHAGEN,  ELSINEUR. 

38.  DJIDDA. — See  ALEXANDRIA. 

39.  EAST  INDIES.— See  BOMBAY,  CALCUTTA,  MADRAS,  TATTA. 

40.  EGYPT. — See  ALEXANDRIA. 


94  MEASURES,  WEIGHTS,  ETC.  [ART.  IV. 

41.  ELSINEUR. — The  same  as  COPENHAGEN,  except  that  the  rix 
dollar  is  divided  into  4  orts  instead  of  6  marcs. 

42.  GALACZ. — See  CONSTANTINOPLE. 

43.  GENOA. — The  palmo  =  9.725   inches.     The    canna  piccola, 
used   by  tradesmen    and  manufacturers,  =  9  palmi;  the   canna, 
grossa,  used  by  merchants,  =12  palmi ;  the  Custom-House  canna 
=  10  palmi.     The  braccio  =  2$  palmi.     In  dry  measure  1  mina  — 
8  quarte  =  36  gombette  =  3$  bushels  nearly.     Salt  is  sold  by  the 
mondino  of  8  mine.     In  liquid  measure  1  mezzarola  =  2  barilla  = 
200  _ptttte  =  39J  gallons.     The   barilla  of  oil  =  17  gallons.     The 
pound  is  of  two  sorts  ;  the  peso  so«z7e  =  4891£gr.  Troy,  for  weigh 
ing  gold  and  silver,  and  commodities  of  small  bulk, — and  the  peso 
ff rosso,  for  weighing  bulky  articles.     The  cantaro  of  100  Ib.  peso 
grosso  =  761b.  14oz.  Avoir. 

MONEY. — 1  lira  Italiana  =  10Q  centesimi  =  l  French  franc.  The 
lira  was  formerly  divided  into  20  soldi,  and  the  soldo  into  12 
denari.  Sales  of  merchandise  continue  to  be  made,  for  the  most 
part,  in  the  old  currency.  6  old  lire  di  banco  =  5  new  lire  very 
nearly. 

44.  GERMANY. — For  weights  and  measures,  see  BREMEN,  DANT- 
zic,   and  TRIESTE.     The  German  short  mile  =  6859-  yards ;    the 
German  long   mile  =  10126yd. ;    the    Hanover   mile  =  11 55 9yd.  ; 
the  Hessian  mile  =  10547yd. ;  the  mile  of  Saxony  =  9905yd. 

MONEY. — The  gold  ducat  =  $2. 24.  The  florin  or  guilder  =  60 
kreutzers  =  240  pfennings.  By  the  Custom-House  valuation,  the 
florin  of  Nuremburg,  Frankfort,  and  the  Southern  States  of  Ger 
many  =  §0.40;  the  florin  of  Augsburg,  Austria,  and  Bohemia,  = 
§0.485 ;  the  florin  of  St.  Gall  =  §0.4036.  The  thaler  or  rix 
dollar  =•  30  groschen  =  360  pfennings.  The  thaler  of  Saxony, 
Prussia,  and  the  Northern  States  of  Germany  =  §0.69.  The 
guilder  of  Surinam,  Cura9oa,  Essequibo,  and  Demarara,  is 
divided  into  20  stivers  of  12  pfennings  each.  Its  value  is  fluctua 
ting,  but  does  not  differ  materially  from  that  of  the  German 
florin. 

45.  GIBRALTAR. — Weights  and  measures  same  as  in  England, 
except  the  arroba  =  25lb.  Av.,  and  the  fanega  for  grain,  =l|bu. 
Wine  is  sold  by  the  gallon,  100  of  which  =  109. 4  U.  S.  gallons. 

MONEY. — Accounts  are  kept  in  current  dollars,  (pesos,)  divided 
into  8  reals  of  16  quartos  each.  12  reals  currency  make  a  cob,  or 


§15.]  MISCELLANEOUS   TABLE.  95 

hard  dollar,  by  which  goods  are  bought  and  sold ;  and  3  of  these 
reals  =  5  Spanish  reals  vellon. 

46.  GREECE. — See  PATRAS. 

47.  HAMBURGH. — The   Hamburgh   foot  =  11. 289  inches.     The 
Rhineland  foot,  used  by  engineers  and  land  surveyors,  =  12.36in. 
The  Brabant  ell,  used  in  the  measurement  of  piece  goods,  =27.585 
in.     A  ton  of  shipping  =  40c.  ft.     DRY  MEASURES. — 1  stock  =  1$ 
hist  =  3  wisps  =  30  scheffels  —  90  fass  =  180  himtems  =  720  spirits  = 
=  134.4bu.    LIQUID  MEASURES. — 1  fuder  =  6  ahms  =  2±  ankers  or  30 
timers  =  120  viertels=2&Q  stubgens  =  480  kanens  =  QQQ  quartiers  = 
1920  oessels  =  229$ga,l.  U.  S.    A  fass  of  wine  =  4  oxhofts  =  Q  tierces. 
An  oxhoft  or  hogshead  of  French  wine  =  62  to  64  stubgens ;  an 
oxhoft  of  brandy  =  60  stubgens.     A  pipe  of  Spanish  wine  =  96  to 
100  stubgens.   A  tun  of  beer  =  48  stubgens.   A  pipe  of  oil  =  820  Ib. 
Whale  oil  is  sold  by  the  barrel  of  6  steckan  =  &2  gallons  U.  S. 
WEIGHTS. — 1  shippound=2%  centners  =  20  lispounds  —  280  pounds 
=  4480  ounces  =  8960  loths  ;  100  Hamburgh  pounds  =  106.8  Ib.  Av. 
In  estimating  the  carriage  of  goods,  the  shippound  is  reckoned  at 
3801bs.     In  things  sold  by  number,  a  gross  thousand=l20Q  ;  a 
ring  =  2  gross  hundred  =  240 ;    a  small  thousand  =  1000  ;    a  shock 
=  3  steigs  =  60 ;  a  gross  =  12  dozen. 

MONEY. — 1  marc  =  16  sols  or  schillings  lubs3-  =  \§2  pfennings  lubs. 
Accounts  are  also  kept,  particularly  in  exchanges,  in  pounds, 
schillings,  and  pence,  or  grotes  Flemish.  The  pound  consists  of  2£ 
crowns,  3f  thalers,  7J  marcs,  20  schillings  Flem.  or  240  grotes  Flem. 
The  moneys  in  circulation  are  divided  into  banco  and  current 
money.  The  former  consists  of  sums  credited  by  the  bank  to 
those  who  have  deposited  bullion  or  specie,  and  is  worth  an  agio 
or  premium  over  current  money.  This  agio  is  usually  about  23 
per  cent.,  but  is  constantly  varying.  Of  the  coins  in  circulation, 
the  rix  dollar  banco  =  about  $1,  and  the  rix  dollar  current  =  $0.80, 
are  the  most  common.  The  Hamburgh  gold  ducat  =  $2. 07.  The 
marc  banco  =  $0.35,  according  to  the  Custom-House  valuation. 

48.  HAVANA.— 108   varas  =  100   yards.      1  fanega  =  3    bushels 
nearly,  or  100 Ib.  Spanish.     1  quintal  =  4  arrobas  =  101  fib.  Av. 
An  arroba  of  wine  or  spirits  =  4.1  U.  S.  gal.  nearly. 

MONEY.— 1  dollar  =  8  reals  plate  =  20  reals  vellon  =  $1.00.  A 
doubloon  =  $17. 

49.  HAYTI. — See  PORT  AU  PRINCE. 

a  Lubs  is  a  contraction  for  money  of  Lubeck. 


96  MEASURES;  WEIGHTS,  ETC.  [ART.  IV. 

50.  HOLLAND. —See  AMSTERDAM. 

51.  JAPAN. — See  NANGASACKI. 

52.  JAVA. — See  BATAVIA. 

53.  KONIGSBERG. — See  DANTZIC. 

54.  LAGUAYRA. — Weights  and  measures  the  same  as  in  SPAIN, 
•with  the  exception  of  the  British  Imperial  gallon. 

MONEY. — The  currency  consists  of  silver  money,  called  macu- 
qucna.  1  dollar  =  8  reals  =  $0.75.  The  money  is  very  unequal  in 
weight  and  purity. 

55.  LEGHORN.— 1   braccio  =  2Q  soldi  =  60  \quattrini  =  240  denari 
=  22.98  inches.     The  canna  =  4  bracci.     The  Tuscan  mile  =  1808 
yards.     1  barile  =  20  fiaschi  =  40  boccali=  80  mezzette  =  12  U.  S. 
gal.    The  barile  of  oil  =  16  fiaschi  =  8.83  U.  S.  gal. ;  it  weighs  about 
10  Ib.  Av.    A  large  jar  of  oil  contains  30  gallons  ;  a  small  one  15  ; 
and  a  box  with  30  bottles  contains  4gal.     Corn  is  sold  by  the  sack 
or  sacco  =  2.0739bu.  U.  S.     The  pound  is  divided  into  12  ounces, 
96  drachms,  288  denari,  and  6912  grani,  and  is  equal  to  5240gr. 
Troy.     The  quintal  or  centinajo  =  100  pounds  =  74.884  Ib.  Av. ;  but 
in  mercantile  transactions,  on  account  of  tares  and  other  allow 
ances,  it  is  usual  to  estimate  100  Ib.  of  Leghorn  =  77  Ib.  Av.     The 
cantaro  is  generally  150  Ib.,  but  a  cantaro  of  sugar  =  151  Ib. ;  of  oil 
=  88  Ib. ;  of  brandy  =  120  Ib.  ;   of  stock  fish,  and  some  other  arti 
cles  =  160  Ib.     The  rottolo  =  3  Ib. 

MONEY. — Accounts  are  principally  kept  inpezze  di  otto  reali,  (01 
dollars  of  8  reals,)  the  pezza  being  divided  into  20  soldi  or  240 
denari.  The  lira  is  another  money  of  account,  chiefly  used  in 
inferior  transactions,  and  subdivided  like  the  pezza.  1  pezza  =  5$ 
lire.  The  moneys  of  Leghorn  have  two  values,  moneta  buona,  01 
the  effective  money  of  the  place,  and  moneta  lunya,  which  is  worth 
oVj  more  than  moneta  buona.  The  pezza  of  account  =  $0.87  ;  the 
pezza  lunga =§0.9076.  The  Tuscan  scudo  or  crown  =  §1.05.  It 
was  subdivided  like  the  pezza. 

56.  LIMA. — See  CADIZ. 

57.  LISBON. — 1    bran^a=2    varas  =  3^    covados   or   cubits  =  10 
palmes  =  86A  inches.     The  palme  —  2  pes  or  feet.     The  legoa  = 
67GOyd.     In  liquid  measure,!   tonnelada  =  2  pipes  =  52  almudes ; 
1  baril  —  18   almudes;  1  almude  —  2  potes  =  12  canadas  =  48  quar- 
tellos  =  4.37gal.  U.  S.     The  value  of  the  almude  in  different  parts 
of  Portugal  varies  from  4£  to  6|  gallons.     The  principal  dry  meas- 


§  15.]  MISCELLANEOUS   TABLE.  97 

ure  is  the  moyo  =  l?>  fancgcu$  =  QQ  alquieres  =  240  quartos  =  480 
«e^mw  =  23.03bu.  U.  S.  The  alquiere  varies  in  different  parts  of 
Portugal  from  3.07  to  3£  dry  gallons.  1  quintal  =  4  arrobas  =  88 
arrutels  or  pounds  =  176  marcs  =  1408  ounces  =  89.047  Ib.  Av. 

MONEY. — 1  milree  =  10QQ  rees  =  $1.12  in  silver.  The  milree  of 
the  Azores  =  $0.83 £  ;  the  milree  of  Madeira  =  $1.00;  the  milree 
of  Brazil  is  fluctuating  in  value.  In  the  notation  of  accounts,  the 
milrees  are  separated  from  the  rees  by  a  crossed  cypher  (©),  and 
the  milrees  from  the  millions  by  a  colon  (:),  thus,  Rs.  2:7009500 
=.  2700  milrees  and  500  rees.  The  crusado  of  exchange  or  old 
crusado  =  400  rees;  the  new  crusado  =  480  rees;  the  testoon  =  ~LQQ 
rees  ;  the  vintem  =  2Q  rees. 

58.  MADRAS. — The  garce,  corn  measure,  =  80  parahs  =  400  mar- 
ca/s  =  137bu.  U.  S.  nearly.  The  marcal  =  8  Buddies  =  64  ollucks, 
When  grain  is  sold  by  weight,  the  garce  =  9256 Jibs.  Goods 
are  weighed  by  the  candy  of  20  maunds  =  500  Ib.  Av.  1  maund=. 
8  via  =  40  seers  =  320  pollams  =  3200  pagodas.  These  are  the 
weights  adopted  by  the  English,  but  those  used  in  the  Jaghire, 
(the  territory  round  Madras  belonging  to  the  Company,)  and  in 
most  other  parts  of  the  Coromandel  coast,  are  called  the  Malabar 
weights.  They  are  the  gursay  or  aarce  =  2Q  baruays  or  candies  == 
400  manunghs  or  maunds  =  9645 Jib.  Av.  The  maund=8  visay  or 
vis  =  320  pollams  =  3200  varahuns. 

MONEY. — The  East  India  Company  and  European  merchants 
keep  their  accounts  at  12  fanams  the  rupee.  1  pagoda  —  3J  rupees 
—  42  fanams  =  3360  cash.  The  star  (or  current)  pagoda  =  $1.84. 

59.  MALABAR. — See  MADRAS. 

60.  MALACCA. — See  SINGAPORE. 

61.  MALAGA. — The  arroba  or  ca?iZara  =  4.19gal.  U".  S.     The  regu 
lar  pipe  of  Malaga  wine  contains  35  arrobas,  but  is  reckoned  only 
at  34;  a  bota  of  Pedro  Ximenes  wines  =  53 -£  arrobas;  a  bota  of 
oil  is  43,  and  a  pipe  35  arrobas ;  a  cargo,  of  raisins  is  2  baskets, 
or  7  arrobas;  a  cask  contains  as  much,  though  only  called  4 
arrobas.     For  other  measures,  weights,    and  coins,   see  CADIZ. 
Accounts  are  kept  in  reals  of  34  maravedis  vellon. 

62.  MALTA. — 1  canna  =  8palmi  =  2^jd.3-    The  Maltese  foot  =  11$ 
inches.     The  ca/iso  or  measure  for  oil  =  5Jgal.  U.  S.     The  salma 
of  corn,   stricken  measure,  =  8.22bu.  U.  S. ;  heaped  measure  is 

a  This  is  the  allowance  usually  made  by  merchants,  in  converting 
Malta  into  English  measure.  In  reality  1  ca««a  =  8.19inches  ;  1  can- 
tar0=17-Hlb.  Av. 

7 


98  MEASURES,  WEIGHTS,  ETC.  [ART.  IV. 

reckoned  16  per  cent.  more.     The  cantaro  —  WQ  rottoli  or  pounds 
=  3000  oncie  =  175  Ib.  Av.a 

MONEY. — In  1825  British  silver  money  was  introduced  into 
Malta ;  the  Spanish  dollar  being  made  legal  tender  at  4s.  4d  ; 
the  Sicilian  dollar  at  4s.  2d. — and  the  Maltese  scudo  at  Is.  8d. 
The  scudo  =  12  tari=  240  grani  =  $0.40. 

63.  MANILLA. — The   same  as  CADIZ,  except  that  weights  are 
estimated   by  piastres.     16   piastres    are    estimated  =  1    Spanish 
pound,  though  they  are  not  quite  so  much.     1  tale  of  silk  =  11 
piastres    or  ounces;    1    catty  =  22 piastres ;    1   marc   of  silver  =  8 
piastres;    1    tale    of  gold  =  10   piastres;    1  picul  =  lQQ   catties  = 
133 Jib.  Av. 

64.  MAURITIUS. — See  PORT  Louis. 

65.  MECKLENBURG. — See  ROSTOCK. 

66.  MEXICO. — See  CADIZ. 

67.  MOCHA. — The^wz  =  25  inches;  the  land  cowdf=18in. ;  the 
long  iron  covid  =  27in.     1  cuda,  liquid  measure,  =  8  nusscahs=  128 
vakias  =  about  2gal.  U.  S.     Grain  is  measured  by  the  kellah,  40 
of  which  =  1  tomand=  about  1701b.  Av.     1  6aAar  =  15/razefo  =  150 
maunds=4QQ  rottoli  =6000  t>aA:Mw  =  4501b.   Av.     There  is  also  a 
small  maund  of  only  30  vakias;  1  Mocha  bahar=16^  Bombay 
maunds  =  13  Surat  maunds=  15.123  seers. 

MONEY. — The  current  coins  of  the  country  are  carats  and  com- 
massees  ;  1  Spanish  dollar  =  8  Mocha  dollars  =  60  commassees  =  420 
carats. 

68.  MOGADORE. — The   canna  or  cubit  =  21    inches.     The  corn 
measures  are,  for  the  most  part,  similar  to  those  of  Spain.     The 
commercial  pound  is  generally  regulated  by  the  weight   of  20 
Spanish  dollars,  therefore  the  quintal  of  100  Ib.  =  119  Ib.  Av.     The 
market  pound  for  provisions  is  50  per  cent,  heavier. 

MONEY. — 1  nutkeel  or  ducat  =  10  ounces  =  40  blankeels  =  9b()  fluce 
=  $0.75. 

69.  MOLDAVIA. — See  GALACZ. 

70.  MONTEVIDEO. — For  weights  and  measures,  see  CADIZ.     The 
current  coins  are  the  Brazilian  patacon  and  Spanish  dollar.     1 
hard  dollar  =  1J-  current  dollars  =  960  centesimos  or  cents  =  $1.00. 
1  real  =  100  centesimos. 

71.  MOROCCO. — See  MOGADORE. 

»  See  Note  on  Page  97. 


§15.]  MISCELLANEOUS   TABLE.  99 


72.  MUSCAT.  —  1  maund=24  ewcAcw  =  8f  Ib.  Av. 

MONEY.  —  1  mamoody  =  20  goz  =  about  $0.05.  The  coins  in  cir 
culation  are  generally  sold  by  weight. 

73.  NANGASACKI.  —  The  inc  is  about  4  Chinese  cubits,  or  6Jft. 
2J  Japanese  leagues  are  computed  to  be  about  1  Dutch  league. 
The  revenues  are  estimated  by  two  measures  of  rice,  the  man  and 
kolf;  the  former  contains  10000  kolfs,  each  3000  bales  or  bags  of 
rice.     The  picul  =  100  catties  =  1600  tads  —  16000   mace  =  160000 
candarines  =  about  130  Ib.  Av.    It  is,  however,  generally  estimated 
at  133Jlb. 

MONEY.  —  Accounts  are  kept  in  taels,  mace,  and  candarines. 
The  Dutch  reckon  the  tael  at  3£  florins,  or  $1.40.  The  coins  in 
circulation,  are  the  old  and  new  it  jib,  and  cobangs  or  copangs,  of 
gold,  —  the  nandiogin,  itaganne,  and  kodama,  of  silver,  —  and  the  seni, 
of  copper,  brass,  and  iron.  Most  of  them  are  without  any  deter 
mined  value,  and  are  therefore  always  weighed  by  the  merchants. 
The  schuit  is  a  silver  piece,  lloz.  fine,  weighing  4oz.  18dwt.  16gr. 
Troy. 

74.  NAPLES.  —  1  canna  =  8  palmi=  96  onzie  =  6ft.  llin.     1  salma 
of  oil  =  16  sto/e  =  256    quarti=1536   mismette.     At   Naples,    the 
salma  =  42f  gal.  U.  S.  ;  at  Gallipoli  it  is  from  3  to  4  per  cent,  less  ; 
at  Bari  it  is  a  little  larger.     The  carro  of  wine  =  2  botti  or  pipes 
=  24  barili  =  1440  caraffe  =  264gal.  U.  S.     The  carro  of  corn  =  36 
fo?noft=52.2bu.  U.  S.     The  cantaro  grosso  =  100  rotloU=I96%lb. 
Av.     The  cantaro  piccolo  =  106  Ib.  Av. 

MONEY.  —  I  ducato  di  regno  =  10  carlini  =  100  grant  =  $0.80.  The 
scudo  of  12  carlini  =  $0.95. 

75.  NORWAY.  —  See  CHRISTIANIA. 

76.  PALERMO.  —  The  yard  or  canna  =  8  palmi=3lyd.  U.  S.  The 
tonna  of  liquids  =  2  ce$m  —  4  barili  =8  quartaro  =  l6Q  quartucci 
=  9fgal.  U.  S.     The  salma  grossa  =  9.48bu.  ;  the  salma  generale  = 
7.62bu.     The  cantaro  grosso  =  100  rottoli  grossi  of  33  onzie,  or  110 
rottoli  sottili  of  30  onzie.     The  cantaro  sottile  =  100  rottoli  sottili,  or 
250  Ib.  of  12  onzie.     The  rottolo  grosso  =  1.93  Ib.  Av.  ;  the  rottolo 
sottile  =  1.751b.  Av.  ;  100  Sicilian  pounds  of  12oz.  =  70  Ib.  Av. 

MONEY.  —  1  ducato  =  10  piccioli  =100  bajocchi  =  $0.80.  Accounts 
are  generally  kept  in  oncie,  tari,  and  grani.  1  oncia  =  3Q  tari  = 
600  grani  =  3  ducati. 

77.  PAPAL  STATES.  —  See  CIVITA  VECCHIA. 


100  MEASURES,  WEIGHTS,  ETC.  [ART.  IV. 

78.  PATRAS. — The  long  pic,  for  measuring  linens  and  woollens, 
=  27in.     The  short  pic,  for  measuring  silks,  =  25in.     The  staro 
of  corn  =  2£bu.  U.  S.     The  quintal  is  divided  into  44  okes,  or 
132  Ib.     100  Ib.  of  Patras  =881b.  Av.     Silk  weight  is  J  heavier. 

MONEY. — 1  phoenix  or  drachmt  =100  Icpta.  The  phoenix  is  a 
silver  coin,  which  should  contain  y9^  of  pure  metal,  and  be  worth 
about  16  cents.  The  lepton  is  a  copper  coin.  The  coinage  has 
been  greatly  debased. 

79.  PERSIA. — See  BUSHIRE. 

80.  PERU. —  See  LIMA. 

81.  PETERSBURG. — 1  sashen  or  fathom  =3  arsheens=48  wershok 
=  7ft.     100  Russian  feet  =  114|  Eng.  ft.     The  verst,  or  Russian 
mile  =  500  sashen  =  5fur.  12r.     The  Polish  short  mile  =  6075yd. ; 
the  Polish  long  mile  =  8101yd.     The  English  inch  and  foot  are 
used  throughout  Russia,  chiefly,  however,  in  measuring  timber. 
In  liquid  measure,  1  sorokovy  =  40  wedros  =  320  krashkas  =  3520 
teharkys=  loOgal.    U.    S.     1  pipe  =  2    oxhofts  =  12    ankers  =  36 
u-edros  =  480  bottles.     1  chetwert  of  corn  =  2  osmins  =  4  pajocks  = 
8    chetwericks  =  64   garnitz  =  5.952bu.    U.   S.     1    berkovitz  =  10 
poods  =  400  pounds  =  12800    loths  =  38400  zolotnicks  =3601b. 
Av.a 

MONEY. — Accounts  are  kept  in  bank  roubles  of  100  copecks.  The 
silver  rouble  =$0.75,  and  was  declared,  by  a  ukase  issued  in 
1829,  to  be  worth  360  copecks,  but  the  value  of  the  paper  rouble 
fluctuates  with  the  exchange.  At  the  Custom-House,  it  is  esti 
mated  at  $0.214. 

82.  PHILIPPINE  ISLANDS. — See  MANILLA. 

83.  PORT  AU  PRINCE. — The  measures  are  the  same  as  in  FRANCE. 
They  are  divided  as  in  Avoirdupois   and  Apothecaries'  weights, 
but  are  about  8  per  cent,  heavier.     The  value  of  the  dollar  is 
about  $0.33,  but  is  constantly  fluctuating. 

84.  PORT  Louis. — The   measures   and  weights    are   those   of 
FRANCE,  previous  to  the  Revolution.     100  Ib.  Fr.  =1081b.  Eng. ; 
16  Fr.  ft.  =  15  Eng.  ft.     The  commercial  velte  =  2  gallons. 

MONEY. — Government  accounts  are  kept  in  sterling  money,  the 
franc  being  received  for  10d.,  and  the  Spanish  dollar  for  4s.  4d. 

a  The  pood  is  reckoned  by  merchants  at  36  Ib.  100  Ib.  Russian  = 
90.26  Ib.  Av.,  according  to  Dr.  Kelly,  or  90.19  Ib.  according  to  Nelken- 
brecher. 


§15.]  MISCELLANEOUS   TABLE.  101 

Merchants  keep  their  accounts  in  dollars  and  cents,  or  in  dollars 
livres,  and  sous. 

85.  PORTO  Rico. — See  HAVANA. 

86.  PORTUGAL. — See  LISBON. 

87.  PRUSSIA. — See  DANTZIC. 

88.  QUEBEC.— The  Paris  foot  is  used  for  all  measures  of  lands 
granted  previous  to  the  conquest,  and  all  measures  of  length, 
unless  a  contrary  agreement   is   made.     The   English   foot,  for 
measuring  lands  granted  since  the  conquest,  and  whenever  spe 
cially  agreed  upon.     The  English  yard  for  cloth  measure.     The 
English  ell  of  5qr.  when  specially  agreed  upon.    The  Canada  minot 
=  l$bu.  for  dry  measure,  except  when  it  is  specially  agreed  that 
the  Winchester  bushel  shall  be  used.     The  Eng.  imp.  gallon  is 
used  for  liquids.     Weights  and  currency,  as  in  England.     The 
pound  =  $4.00. 

89.  RANGOON. — I  ten  or  basket  =  4  suits  =  8  sarots  =  16  pyis  = 
64  salts  =  128  lamts  =  256  lamyets.     A  ten  of  clean  rice  ought  to 
weigh  16  vis  or  58.4  Ib.  Av.     1  paiktha  or  vis  =  100  Jcyats  or  ticals 

=  400  mat' hs  =  800  TOMS  =  1600  6aw  =  6400  large  rices  =  12800 
small  rwts  =  3.65  Ib.  Av. 

MONEY.— Lead  is  used  for  small  payments;  gold  and  silver, 
(principally  the  latter,)  for  larger  ones.  There  are  no  coins,  but 
the  metal  must  be  weighed,  and,  very  generally,  assayed  at  every 
payment.  Every  new  assay  of  silver  costs  the  owner  2£  per  cent. 

90.  RIGA.— 1   clafter  =  Z  ells  =  6  feet  =  64.74in.     1  fuder,  for 
liquids,  =  6  ahms  =  24  ankers  =  120  quarts  =  720  stoofs  =  248gal. 

J.  S.  The  loo/for  grain  =  1.9375bu.  A  last  =  4S  loofs  of  wheat, 
barley,  or  linseed,— 45  loofs  of  rye,— or  60  loofs  of  oats,  malt,  or 
beans.  1  shippound  =  20  lispounds  =  400  pounds  =  800  marcs  = 
12800  loths  =  368.68 Ib.  Av. 

MONEY.— See  PETERSBURG.  The  current  rix  dollar  of  Riga  = 
90  groschen  =  $0.69. 

91.  ROSTOCK.— 1  ell  =  2  feet=22.7Q  Eng.  in.     The  last  =  96 
scheffcls  =  116  imp.  bu.  of  oats,  or  104  imp.  bu.  of  other  grain. 
The  commercial  weights    are    the    same  as  those   of  Hamburg. 
There  are  other  weights,  5  per  cent,  heavier  than  these,  which 
are  principally  used  in  the  trade  with  Russia. 

MONEY.— The  rix  dollar,  new,  =  32  schillings,  and  contains 
UU.lgr.  pure  silver. 


102  MEASURES,  WEIGHTS,  ETC.  [ART.  IV. 

92.  RUSSIA. — See  PETERSBURG  and  RIGA. 

93.  SALONICA. — Weights  and  measures  same  as  at  Constanti 
nople,  except  the  kisloz  or  killow,  which  =  3.78  kisloz  of  Smyrna. 

MONEY. — Accounts  are  kept  in  piastres  of  40  paras,  or  120  aspere. 
The  coins  are  those  of  Constantinople. 

94.  SARDINIA. — See  CAGLIARI  and  GENOA. 

95.  SIAM. — See  BANGKOK. 

96.  SICILY. — See  PALERMO. 

97.  SINDE. — See  TATTA. 

98.  SINGAPORE. — English  weights  and  measures  are  frequently 
used  in  reference  to  European  commodities.     Piece  goods  and 
many  other  articles,  are  sold  by  the  corge  or  score.     A  coyan  of 
rice  or  salt  =  40  piculs.     Nearly  everything  is  sold  by  weight,  as 
in  China.     The  picul  =  100  catties  —  133£  Ib.  Av.     Gold  dust  is 
sold  by  a  Malay  weight,  called  the  bunakal,  —  832gr.  Troy.    Bengal 
rice,  wheat,  and  pulses,  are  sold  by  the  bag,  containing  2  Bengal 
maunds,  or  164 Jib.  Av. 

MONEY. — Merchants'  accounts  are  kept  in  Spanish  dollars, 
divided  into  100  parts,  represented  either  by  Dutch  doits  or  by 
English  copper  coins  of  the  same  value. 

99.  SMYRNA. — See  CONSTANTINOPLE.  The  kisloz  =  1.456  bushels. 

100.  SOUTH  AFRICA. — See  CAPE  TOWN. 

101.  SPAIN. — See  ALICANT,  BARCELONA,  CADIZ,  GIBRALTAR,  and 
MALAGA.      Vellon  is  the  old  copper  coin  of  Castile,  and  is  of  but 
half  the  value  of  the  plate  or  silver  currency. 

102.  STOCKHOLM.— The  rod  =  2f  fathoms  =  8  ells  or  alnas  =  16 
feet  =  15.58  Eng.  ft.     1  pipe  =  2  oxhofts  =  3  ohms  =  6  eimers  =  12 
ankers  =  180  kannor  =  360  stup  =  124Jgal.  TJ.  S.     In  corn  meas 
ure  1  tun  or  barrel  =  2  spann  =  8  quarts  =  32  kappor  =  4^bu.  U. 
S.     The  victual!  or  commercial  weights  are  1  skippund  =  20  lis- 
punds  =  400  punds  =  375  Ib.  Av.    The  iron  weights  are  four-fifths 
of  the  victual!.     1  iron  skippund  =  20  mark  punds  =  400  marks  = 
300  Ib.  Av. 

MONEY. — For  many  years,  there  were  no  coins  except  copper  in 
circulation,  but  both  silver  and  gold  are  now  coined.  1  rix  dollar 
—  48  skillings  =  576  rundstycks  or  'ore.  The  banco  currency  is 
worth  30  per  cent,  of  the  silver  currency.  A  rix  dollar  banco  is 
•worth  about  $0.40.  A  silver  rix  dollar  =  $1.06  =  2  dollars  32 
skillings  banco. 


§15.]  MISCELLANEOUS    TABLE.  103 

103.  SWEDEN. — See  STOCKHOLM. 

104.  TATTA. — 1  guz  —  16  garces  =  32  inches.     But  1  guz  of 
cloth,  at  Tatta  =  34  inches.     1  carval  of  wheat  =  60  cossas  =  240 
twiers  =  960  puttoes  =  22  Pucca  maunds,  or  21  Bombay  parahs. 
The  gross  weights  are  1  maund  =  40  seers  =  640  annas  —  2560 
pice,  =  74  Ib.  5oz.  7dr.  Av.     The  small  weights,  1  tolah  =  12  mas- 
sas  =  72  ruttees  =  576  hubbahs  —  1728  moons  =  about  6  dwt.  Troy. 

MONEY. — 1  rupee  =  50  carivals  =  600  pice  =  28800  cowries  = 
$0.55. 

105.  TRIESTE. — The  Bohemian  mile  =  10137  Eng.  yd. ;  the  Hun 
garian  mile  =9113  Eng.  yd.     The  ell  for  woollens  =  26.6in. ;  the 
ell  for  silk  -—  25.2in.    .The  orna  or  rimer,  liquid  measure,  =  40 
boccali  =  about  15gal.  U.  S.     The  barile  =  173^gal.  U.  S.     The 
orna  of  oil  =  5£  caffisi=  17gal.  U.  S.     The  principal  dry  meas 
ure  is  the  stajo  or  staro  =  2.34bu.  U.  S.     Sometimes  the  Vienna 
metzen  is  used,  =  2  polonicks  =  1.728bu.     The  commercial  pound 
=  4  quarters  —  16  ounces  =  32  loths  =  8639gr.  Troy. 

MONEY.  —  Contracts  are  usually  in  silver ;  gold  coins,  not 
being  legal  tender,  pass  only  as  merchandise.  Mercantile  ac 
counts  are  usually  kept  in  convention  money,  so  called  from  an 
agreement  made  by  some  of  the  German  princes  in  1763.  The 
current  coins  are  dollars,  half-dollars  or  florins,  and  zivanzigers,  or 
pieces  of  20  kreutzers.  Ten  dollars  are  coined  out  of  the  Cologne 
marc  =  3608gr.  Troy,  of  pure  silver.  The  florin  =  60  kreutzers 
=  240  pfennings  =  $0.48£. 

106.  TUNIS. — The  pic  for  woollens  =  26. Sin. ;    the  silk  pic  = 
24. Sin. ;  the  linen  pic  =  18.6in.     The  principal  oil  measure  is  the 
metal  or  mettar  =  about  5-Jgal.,  but  it  is  of  different  dimensions  in 
different  parts  of  the  country.     The  wine  measure  is  the  millerolle 
of  Marseilles  =  6  J   mitres  =  14.1    imp.  gal.     The   principal    corn 
measure  is  the  ccr/zz  =  16  whibas  =  I32  sahas  =  14%  imp.  bu.     The 
cantaro  =  100  rottoli  or  pounds  =  111. 05  Ib.  Av.     Gold,  silver,  and 
pearls  are  weighed  by  the  ounce  of  8  meticols  ;  16  of  these  ounces 
make  the  Tunis  pound  =  7773. 5gr.  Troy. 

MONEY. — 1  piastre  =  16  carobas=52  aspers  =  624c  burbine  = 
$0.24.  The  piastres  coined  since  1828  are  worth  but  $0.125. 

107.  TURKEY. — See  CONSTANTINOPLE  and  SALONICA. 

108.  TUSCANY. — See  LEGHORN. 
109    URUGUAY. — See  MONTEVIDEO. 


104  MEASURES,  WEIGHTS,  ETC.  [ART.  IV. 

110.  VALPARAISO.—  The  vara  =  33.  384in.     The  fanega,  or  prin 
cipal  corn  measure,  contains  3439  c.  in.     The  quintal  =  4  arrobas 
«=  100j90ttrafo=101.441b.  Av.     See  CADIZ. 

111.  VENEZUELA.—  See  LAGUAYRA. 

112.  VENICE.—  The  woollen  Araccz'o  =  26.6in.  ;  the  silk  braccio 
=  24.8in.  ;  the  Venice  foot  =  13.  68in.     The  botta  =  2  migliaje  =  5 
bigonzi  =  80  miri.     The  miro  of  oil  =  4.028gal.  U.  S.,  or  25  Ib.  peso 
grosso.     The  anfora  of  wine  —  4  bigonzi  =  8  mastelli  =  48  secchii  — 
192  bozze  =  76S  quartuzzi  =  137  g&l.  U.  S.     The  moggio,  for  corn, 
=  4sto/e  =  16  quarte  =  Q4  quartaroli  —  9.08bu.  U.  S.     100  Ib.  peso 
gtosso  =  105.186  Ib.  Av.  ;  100  Ib.  peso  sottile  =  66.428  Ib.  Av.     The 
libra  Ilaliana  is  sometimes  used,  and  is  equivalent  to  the  French 
kilogramme. 

MONEY.  —  -1  lira  Italiana=-I00  centestmt  =  1000  millesimi.  The 
lira  is  supposed  to  be  of  the  same  value  as  the  franc,  but  the  lire 
actually  in  circulation,  are  worth  only  about  $0.09.  At  the  Cus- 
tom-House,  the  lira  is  estimated  at  $0.16. 

1  6.    ANCIENT  MEASURES,  WEIGHTS,  AND  COINS.* 

1.  SCRIPTURE  LONG  MEASURE.  —  1  schoenus  =  lO  Arabian  poles  = 
13J  EzekieVs  reeds  =  20  fathoms  =  80  cubits  =  160  spans  =  480  palms 
=  1920  digits  =  46.2    yards.     A  day's  journey  =  8  parasangs  =  24 
eastern  miles  =  48  Sabbath-day's  journeys  =  240  stadia  =  96000  cubits 
=  33.264  miles.    LIQUID  MEASURE.  —  1  coron  or  chomer=~\.0  bath  or 
epha  =  30  scah  =  QO  Am  =  180  cab  =  720  Zo^  =  9GO  capA  =  103.35 
gallons.     DRY  MEASURE.  —  1  coron  or  cAo?ner  =  2  latech  =  10  epha 
=  30  *ea/i  =  100  Corner  =  180  ca6  =  3600  aachal  =  10.9616  bushels. 
WEIGHTS.  —  1  talent  =  50  maneh  =  3000  shekel  =  111\\).  Troy,  nearly. 
MONEY.—  1  taZercf  =  60  maneh  =  3000  shekels  =  6000  6e£aA  =  60000 
gerah  =  £342  3s.  9d.    A  talent  of  gold  =£5475.    The  solidus  aureus 
or  ««eto2a  =  12s.  Od.  2qr.     The  siclus  aureus  =  £l  16s.  6d. 

2.  GRECIAN  LONG  MEASURE.  —  1  milion  or  mile  =  8  stadios,  dulos, 
or  furlongs  =  800  orgy  a  or  paces  =  3200  pechys  or  larger  cubits  = 
3840  pygon  =  42G6%  pyame  or  cubits  =  4800  pous  or  feet  =  6400 
spithame  =  6981  T9r    orthodoron  =  7680   rficAajy  =  19200   rforon   or 
dochme  =  76800  dactylos  or  digits  =  4835  feet.    LIQUID  MEASURE.  — 
1  metretes  =  12  chous  =  72  xe«^5  =  144  cotyle  =.  576  oxybaphon  =  864 


*  McCulloch,  Gregory,  Brande,  Lavoisne,  Enc.  Amer. 


§16.]  MISCELLANEOUS   TABLE.  105 

cyathos  —  1728  conche  =  3456  mystron  —  4320  cheme  =  8640  cochlia- 
rion  — 10.335  gallons.  DRY  MEASURE. — 1  medimnos  =  4S  chocnix  = 
72  xestes  =  14-1  cotyle  =  576  oxybaphon=SQ4i  cyathos  =  8640  coch- 
liarion  =  1.0906  bushels.  WEIGHTS. — 1  talent  =  60  mince  =  6000 
drachma  =  36000  obolos  =  56  Ib.  Av.  MONEY. — 1  tetradrachma  or 
stater  =  2  didrachma  =  4  drachma  =  6  tetrobolon  =  12  diobolon  = 
24  o£ota,9==48  hemiobolos  =  96  dichalcos  =  192  c/mteos  ==  1344 
lepton  =  2s.  7d.  The  pentadrachma  =  5  drachma.  The  drachma 
and  its  multiples,  were  of  silver, — the  other  coins,  mostly  of  brass. 
The  stater  aureus  =  25  Attic  drachmas  of  silver.  The  stater  (7?/zi- 
cenus,  stater  Philippicus,  and  stater  Alexandrinus  =  28  silver 
drachmas.  The  stater  Daricus  and  stater  Crccsius  =  50  silver 
drachmas. 

3.  ROMAN  LONG  MEASURE. — 1  milliare  =  8  stadia  =  1000  passus 
=  2000  yracftw  =  3333 |  cwfo'te  =  4000  palmipedes  =  5000  jsec/es  or 
feet  =  20000  palmi  minores  =  60000  uncice  =  80000  %zft'  transversi 
=  4835  feet.     SQUARE  MEASURE. — 1  #5  =  !^  dcunx  =  \\   dextans 
=  1£  dodrans  =  l%  bes=^l^  septunx  =  2  semis  or  actas  major  =  2* 
quincunx  =  3  triens  =  4  quadrans  =  6  sextans  or  actas  minimus  =•  8 
dwraa  or  sescM/zcm  =  12  M7zcm  =  28800sq.  feet.     LIQUID  MEASURE. — 
1  culeus  =  20  amphora  =  40  wrn«  =  160  co/iyzV  =  960  sextarii  =  1920 
/iemmoe  =  3840    quartarii  =  768Q  acetabula  — 11520  cyathi=  46080 
ligulce  — 143.4258  gallons.    DRY  MEASURE. — 1  modius  =  2  semimodii 
=  16  sextarii  =  32  hemince  =  128  acetabula  =  192  cyathi—768  ligulm 
=  1.0141  pecks.     WEIGHTS. — 1  libra  =  12  uncice  =  36  rfMeWce  =  48 
mct'tici  =  72    sextulce  =  5246gr.    Troy.     MONEY. — 1    denarius  =  2 
quinarii  or  victor iati  =  4  sestertii  =  10  libellce  or  asses  =  20  sembellce 
=  40  teruncii  =  7d.  3qr.     All  above  the  as,  (and  sometimes  the 
as  also,)  were  of  silver,  the  rest  of  brass.     The  gold  coin  was  the 
aureus,  which   generally  weighed    double  the  denarius,  and  was 
about  equivalent  to  18  denarii  in  value. 

4.  VARIOUS  ANCIENT  MEASURES. — The   Olympic  stadium  =  202 
yards;  the  Alexandrian  stadium  =  110yd.  ;  the  Egyptian  stadium 
=  245yd.  ;  the  stadium  of  Aristotle  =  115yd.  ;  the  Persian  par as ang 
=  6440yd.  ;  the  Egyptian  schccne  =  11120yd.  ;  the  ancient  Spanish 
mile  =  1374yd.  ;  the  ancient  British  mile  =  148yd.  ;  the  league  of 
Gaul  =  2440yd.  ;  the  rasta  of  Germany  =  4705yd. 

[These  measures  are  given  principally  on   the   authority  of 
Lavoisne.] 


106  MEASURES,  WEIGHTS,  ETC.  [ART.  IV. 

§  IT1.     EXAMPLES  FOR  THE  PUPIL 

1.  The  great  bell  of  Moscow  was  cast  in  1653,  during 
the  reign  of  the  Empress  Anne.     Its  weight  is  estimated  at 
198tons  2cwt.  lqr.a     To  what  would  this  be  equivalent  in 
Troy  weight  ? 

2.  In  the  first  year  after  the  discovery  of  gold  in  Califor 
nia,  the  amount  collected  is  supposed  to  have  been  about 
$5000000.    Estimating  its  average  value  at  $16  per  oz.  Troy, 
what  would  be  the  weight  of  the  whole  in  Avoirdupois  ? 

3.  Find  the  weight  in  Troy  grains  of  the  French  twenty- 
franc  piece  of  gold,  and  also  of  the  silver  five-franc  piece. 

4.  The  distance  from  Boston  to  LiverpooP  is  about  2883 
statute  miles.     At  the  average  rate  of  8  knots  an  hour,  in 
what  time  would  a  vessel  sail  from  one  of  these  places  to 
the  other  ? 

5-10.  On  the  Cathedral  of  Paris  a  bell  was  placed  in 
1680,  which  weighed  340cwt. ;  a  bell  was  cast  in  Vienna, 
in  1711,  weighing  354cwt. ;  in  Oliniitz  is  one  of  358cwt. ; 
the  Smanne,  a  fine-toned  bell  at  Erfurt,  which  has  a  large 
proportion  of  silver  in  its  composition,  weighs  275cwt.,  and 
has  a  clapper  weighing  llcwt.a  Reduce  each  of  these 
weights  to  kilogrammes,  and  find  the  amount  of  the  whole. 
Ans.  for  the  amount,  67966.308kil. 

11-15.  Great  Tom,  of  Christ  Church,  Oxford,  weighs 
17000  Ib. ;  Great  Tom  of  Lincoln,  9894  lb. ;  the  bell  of  St. 
Paul's,  London,  8400  lb. ;  a  bell  at  Nankin,  China,  is  said 
to  weigh  50000  lb. ;  and  seven  at  Pekin,  120000  lb.  each.* 
Required  the  weight  of  each  in  Amsterdam  ponds. 

16-25.  Reduce  the  span  and  height  of  each  of  the  fol 
lowing  bridges,0  to  the  measure  required  in  the  right-hand 
column. 

a  Enc.  Amer.  b  Cunard  Steamers.  c  Brande's  Enc. 


§17.] 


MISCELLANEOUS   EXAMPLES. 


107 


Name  of  Bridge. 

Material           River.                   Place. 

Wi  lest 

Arch. 
Heijht. 

Bate. 

Measure. 

Colossus  

Wood  Schuylkill  j  Philadelphia  340  ft.: 

20    ft. 

1813 

Tur'h  piks. 

Piscataqua  .  .  . 
Southwark  .  .  . 

"     iPiscutaqua  Portsmouth   J250 
Iron  Thames       ;  London            240 

24>3/' 

1794 

1818 

Swedish  ft. 
Arns.  palms. 

Sunderland.  .  . 

"     iWear            Sunderland    |240 

30     ' 

1796 

Prussian  ft. 

Bamberg  
Trenton  

WoodiResfnitz        Germany 
"      Delaware     New  Jersey 

208 
200 

32^' 

1809 

1801 

Bremen  ft. 
Venice  ft. 

Vielle  Brioude 

Stone  Allier            Brioude 

183^ 

70^;' 

1454 

Fr.  metres. 

Ulm  

"      Danube        Ulm 

181  "y 

22^' 

1  80(1 

Dantzic  ft. 

Waterloo  .... 

"     .Thames       .London 

120 

32     ' 

1816 

Script,  cub. 

Blackfriars  .  .  . 

"     [Thames       1  London 

100 

41^' 

1771 

Russian  ft. 

26.  Reduce  2  tons  13cwt.  3qr.  171b.  to  Canton  weight. 

27.  Reduce  159  taels  3  mace  5  candarines   7  casli  to 
Troy  ounces.  Ans.  193.663oz. 

28.  In  some  places  milk  is  sold  both  by  beer  measure 
and  by  wine  measure.     To  how  many  wine  gallons  would 
the  difference  amount  in  a  year,  in  a  family  that  takes  2qt. 
per  day  ? 

29-38.  At  84.84  per  pound  sterling,  what  is  the  cost  per 
c.  ft.  in  English  money,  of  each  of  the  following  public 
buildings  ? 


U  S.  Public  Buildings. 

Contents-^ 

Cost.b 

Capitol 

4147400  c 

ft 

^2950000 

Treasury  Dull  din"1 

1044740 

648743 

Patent  Office  

1466060 

417550 

General  Post  Office    

1071252 

4597(55 

Girard  College  

2545485 

1497500 

New  York  Custom-House   .   . 
Boston  Custom-House  .... 
Philadelphia  Custom-House  . 
Trinity  Church,  New  York    . 
Smithsonian  Institution  .   .  . 

906000 
730000 
530613 
821070 
1545000 

960000 
776000 
257452 
338000 
215000 

kJlUll/JLLDUUlOill    J.110LJLLUHUJ.1     .      .      .  lUt-JUUU  Z,1«JUUU 

39.  If  you  had  a  pair  of  scales,  but  no  weights,  how 
would  you  weigh  31b.  7oz.  by  using  water? 

40.  How  many  cents  weigh  a  pound  Avoirdupois  ?     How 
many  eagles  ?     How  many  half-dollars  ? 

a  This  was  the  largest  single  span  in  the  world.  The  bridge  wag 
destroyed  by  an  incendiary,  September  1,  1838,  and  a  wire  suspension 
bridge  has  since  been  erected  in  its  place. 

b  R.  Dale  Owen. 


108  MEASURES,  WEIGHTS,  ETC.  [ART.  IV. 

41.  At  80.186  per  franc,  what  would  be  the  value  in 
francs,  of  a  diamond  weighing  15  carats,  at  $36  for  the  first 
carat  ?  Ans.  43548fr.  40c.a 

42.  What  is  the  value  in  Federal  Money  of  the  French 
twenty-franc  piece,  provided  the  gold  is  of  the  present  mint 
standard,  and  of  full  weight  ? 

43-44.  The  largest  known  diamond  was  found  in  Gol- 
conda,  in  1550,  and  is  in  the  possession  of  the  Great  Mogul. 
It  is  half  the  size  of  a  hen's  egg,  and  is  said  to  weigh  900 
carats. b  In  the  usual  mode  of  estimating  diamonds,  what 
would  be  its  value,  at  $9  for  the  first  carat  ?  What  is  its 
weight  in  ounces  Avoirdupois  ? 

45-50.  The  Greeks  reckoned  by  the  era  of  the  Olympiad, 
which  began  at  the  summer  solstice,  776  B.  C.  The  Roman 
era  commenced  with  the  building  of  the  city,  April  24, 
B.  C.  753.  The  Julian  era  dates  from  the  reformation  of 
the  calendar  by  Julius  Caesar,  B.  C.  45.  The  Mohammedan 
era  dates  from  the  Hegira,  July  16,  A.  D.  622.  The  era 
of  Sulwanah,  used  in  a  great  part  of  India,  corresponds  with 
A.  D.  78.  The  era  of  Yezdegird,  used  in  Persia,  began 
June  16th,  A.  D.  632.b  Find  the  time  that  elapsed  between 
each  of  these  dates  and  the  commencement  of  the  Jewish 
era,  3760  B.  C. 

51-67.  Find  the  value  of  100  Ib.  Av.  in  the  commercial 
weights  of  each  of  the  following  places :  Alexandria, 
Amsterdam,  Bombay,  Bremen,  Cadiz,  Calcutta,  Canton, 
Civita  Vecchia,  Constantinople,  Hamburg,  Madras,  Naples, 
Patras,  Petersburg,  Stockholm,  Trieste,  Venice. 

68.  Newly  burned  charcoal  will  absorb  90  times  its  bulk 
of  ammonia,  or  35  times  its  bulk  of  carbonic  acid.c  How 
many  cubic  inches  of  each  of  these  gases  jWOuld  be  absorbed 
by  17  cubic  feet  of  new  charcoal  ? 

a  No  fraction  of  a  franc  is  counted,  less  than  5  centimes.     For  the 
method  of  estimating  the  value  of  diamonds,  see  $  12,  p.  73. 
bBrande.  c  Carpenter. 


§  17.]  MISCELLANEOUS   EXAMPLES.  109 

69.  A  spider's  thread  is  in  some  instances  no  more  than 
Woo  °f  an  *ncn  in  diameter.     Each  thread  is  formed   of 
from  4  to  6  filaments,  and  each  filament  of  not  less  than 
1000  fibrils. a     How  many  fibrils  would  occupy  a  foot  in 
breadth,  allowing  1100  to  a  filament,  5  filaments  to  a  thread, 
and  3000  threads  to  an  inch  ? 

70.  A  large  sunflower  has  been  observed  to  lose  1  Ib.  4oz. 
during  24  hours  by  evaporation,  and  a  cabbage  lost  1  Ib.  3oz. 
during  the  same  time.b     At  this  rate,  in  what  length  of 
time  would  the  evaporation  from  each  plant  amount  to  Icwt.  ? 

Ans.  Sunflower,  89 d.  14 fh. 
Cabbage,    94d.    7-}^h. 

71.  Find  the  value  of  each  of  the  French  weights  and 
measures,  in  the  weights  and  measures  of  the  United  States. 

72-81.  Only  4  of  the  55  chemical  elements a  seem  to  be 
essential  to  the  constitution  of  living  matter,  viz  :  Carbon, 
Hydrogen,  Oxygen,  and  Nitrogen.  What  weight  of  each 
element  is  contained  in  Icwt.  of  each  of  the  following  sub 
stances,  the  weight  of  1  part  of  Carbon  being  represented 
by  6,  1  part  of  Hydrogen  by  1,  1  part  of  Oxygen  by  8, 
and  1  part  of  Nitrogen  by  14  ? 


Gum. 

Sugar. 

Starch. 

Lignin. 

Wax. 

Carbon, 

414 

parts. 

421 

428 

500 

806 

Hydrogen 

,     65 

« 

64 

63 

56 

114 

Oxygen, 

521 

a 

515 

509 

444 

80 

1000 

(( 

1000 

1000 

1000 

1000 

Gluten 

Gelatin. 

Albumen. 

Fibrin, 

Fat. 

Carbon, 

557 

parts. 

483 

516 

520 

790 

Hydrogen 

,     78 

« 

80 

75 

72 

118 

Oxygen, 

220 

a 

276 

259 

250 

92 

Nitrogen, 

145 

a 

161 

150 

158 

00 

1000 

it 

1000 

1000 

1000 

1000 

a  Carpenter. 

b  Griffiths. 

HO  THE   FARM.  [ART.  V. 

82.  Human  hair  is  from  3-J0  to  rj<j  of  an  inch  in  diame 
ter.      How  many  hair's  breadths  in  49  yards,  if  there  are 
370  in  an  inch  ? 

83.  Estimating  the  entire  population   of  the  world  at 
1000000000,  what  would  be  the  size  of  a  field  that  would 
hold  the  whole,  if  each  person  occupied  1  sq.  yd.  ? 

Ans.  322  sq.  m.  531A.  2760  sq.  yd. 

84.  The  average  weight  of  the  lignin,  or  woody  fibre,  of 
an  oak,  has  been  estimated  at  60  tons,  30  tons  of  which  are 
carbon. a    According  to  this  estimate,  how  many  square  miles 
of  oak  forest  would  yield  an  amount  of  carbon  equivalent  to 
that  contained  in  the  atmosphere,  supposing  the  oaks  to 
stand  1  rod  apart, — carbonic  acid  consisting  of  T3r  carbon, 
and  -ff  oxygen  ?b 

85.  What  is  the  value  of  the  labor  expended  in  manufac 
turing  1  Ib.  Avoirdupois  of  watch-springs,  each  spring  being 
worth  $2,  and  weighing  -fa  of  a  grain,  and  the  value  of  the 
iron  employed,  being  1  cent  ? 


V.    THE  FARM. 
1 8.     RULES  FOR  DETERMINING  THE  WEIGHT  OF  LIVE 

CATTLE. 

1.  Measure  in  inches,  the  girth  round  the  breast,  just 
behind  the  shoulder-blade,  and  the  length  of  the  back  from 
the  tail  to  the  fore-part  of  the  shoulder-blade.  Multiply 
the  girth  by  the  length  and  divide  by  144.  If  the  girth  is 
less  than  3ft.,  multiply  the  quotient  by  11 ;  if  between  3ft. 
and  5ft.,  multiply  by  16;  if  between  5ft.  and  7ft,,  multiply 
by  23 ;  if  between  7ft.  and  9ft.,  multiply  by  31.c  If  the 
animal  is  lean,  deduct  ^  from  the  result.  Or, 

»  Carpenter.  b  See  Ex.  17,  Sect.  6. 

c  Chambers's  Information  for  the  People,— Brit.  Husbandry. 


§19.]  MEASUREMENT   OP   GRAIN.  Ill 

2.  Take  the  girth  and  length  in  feet.  Multiply  the 
square  of  the  girth  by  the  length,  and  multiply  the  product 
by  3.36.  The  result  will  be  the  answer  in  pounds.  Either  of 
these  rules  will  be  found  useful  in  making  approximate  esti 
mates  of  the  weight  of  cattle,  but  experienced  judges  depend 
more  upon  observation  than  upon  arbitrary  rules.  The  live 
weight,  multiplied  by  .605,  gives  a  near  approximation  to 
the  net  weight. 

19.    RULES  FOR  MEASURING  GRAIN. 

1.  If  the  grain  is  heaped  on  a  floor,  in  the  form  of  a 
cone,  measure  the  depth  and  slant  height  in  inches.     Mul 
tiply  the  difference  of  the  squares  of  these  two  measure 
ments  by  .0005  of  the  depth,  for  the  contents  in  bushels. 

2.  If  the  grain  is  heaped  against  the  side  of  a  building,  take 
£  of  the  result  obtained  by  the  first  rule.     If  heaped  against 
an  inner  corner  of  a  building,  take   i, — and  if  against  an 
outer  corner,  take  f  of  the  same  result. 

3.  If  the  grain  is  in  barrels  or  common  casks,  the  quan 
tity  may  be  found  with  tolerable  accuracy  by  the  following 
rule  :  "  To  .4  of  the  square  of  the  bung  diameter,  add  J  of 
the  square  of  the  head  diameter,  and  4  of  the  product  of 
the  two  diameters;    multiply  the  amount  of  these  three 
quantities  by  the  length  of  the  cask,  and  divide  the  product 
by  265  for  wine  gallons,  by  324  for  ale  gallons,  or  by  309 
for  half-pecks."* 

4.  If  the  grain  is  in  a  bin  or  crib,  .8  of  the  number  of 
cubic  feet,  will  give  the  number  of  bushels.     Two  bushels 
of  corn  on  the  cob  will  give  about  l^bu.  when  shelled.     A 
"  barrel"   of  shelled  corn,  (in  the  Southern  States,)  is   5 
bushels.b 


a  Simplified  from  Dr.  Hutton's  rule.    For  more  accurate  rules,  see 
the  article  on  gauging. 
b  Scientific  American. 


112  THE    FARM.  [ART.  V. 

5.  If  the  grain  is  not  of  uniform  depth  or  breadth,  take 
the   dimensions  in  several  different  places,  and  take  the 
average  length,  breadth,  and  height,  with  which  proceed  as 
in  Rule  4. 

6.  To  estimate  the  yield  per  acre,  measure  the  quantity 
produced  on  1  sq.  rod,  and  multiply  by  160,  or  measure  the 
produce  of  11  yards  square,  and  multiply  by  40. 

SO.    WEIGHT  OF  GRAIN  AND  HAY. 

1.  The  standard  weights  of  graina  in  Great  Britain,  and  in 
many  parts  of  the  United  States,  are  as  follows  :    Wheat, 
60  Ib.  per  bushel ;  Indian  corn  and  rye,  56  Ib. ;  barley,  48  Ib. ; 
oats,  32  Ib.     In  Indiana,  a  bushel  of  rye  is  46  Ib.,  and  a 
bushel  of  oats,  33  lb.b    Beans  and  clover  seed  weigh  nearly 
the  same  as  wheat ;  castor  oil  beans,  and  timothy  seed,  the 
same  as  corn ;  potatoes,  40  Ib.  per  bushel;  buckwheat,  52  Ib. ; 
salt,  50  Ib. ;  bran,  20  Ib. ;  dried  peaches,  24  Ib. ;  dried  apples, 
22  Ib. ;  blue  grass  seed,  14  lb.c 

2.  Ten  solid  yards  of  hay  which  has  settled  in  the  stack 
during  winter,  weigh  about  1  ton.     In  stacks  more  than  a 
year  old,  9yd.,  and  sometimes  8yd.  make  a  ton.     Clover 
takes  11  or  12yd.,  and  sometimes  when  it  has  been  stacked 
very  dry,  13yd.  to  a  ton.     In  the  barn,  400  c.  ft.  in  the  bay, 
or  500  c.  ft.  on  the  scaffold,  make  about  1  ton.d     If  a  stack 
of  hay  has  a  circular  base,  its  contents  may  be  found  very 
nearly  by  the  following  rule  :     Measure  the  circumference 
at  the  bottom  of  the  stack,  and  the  height  of  the  stack. 
Multiply  the  square  of  the  circumference  by  the  height,  and 
divide  the  product  by  40.     The  contents  may  also  be  meas 
ured  by  finding  the  average  length,  breadth,  and  height,  and 
forming  the  continued  product  of  the  three  dimensions.6 

a  Brit.  Husbandry.  b  Mer.  Mag.  c  Scientific  Amer. 

d  Colman,  and  private' information. 

e  A  product  that  is  formed  by  multiplying  together  three  or  more 
factors,  is  called  the  "  continued  product"  of  those  factors.  Thus,  the 
continued  product  of  3,  17,  and  5,  is 


§22.]  MISCELLANEOUS  EXAMPLES.  113 

21.    MEASUREMENT  OF  LAND. 

1.  If  the  field  is  square,  or  oblong,  multiply  the  length 
by  the  breadth,  and  the  contents  will  be  the  area.     If  the 
dimensions  are  taken  in  paces,  the  area  will  be  in  square 
yards.     Multiply  the  number  of  square  yards  by  4,  and 
divide  the  product  by  121,  and  you  will  obtain  the  area  in 
square  rods ;  or  divide  the  number  of  square  yards  by  484, 
and  point  off  one  figure  from  the  right  of  the  quotient,  and 
the  result  will  be  the  area  in  acres. 

2.  If  the  field  is  irregular,  divide  it  into  triangles.     In 
each  triangle,  multiply  the  length  of  the  longest  side  by  the 
shortest  distance  from  that  side  to  the  opposite  angle.    Add 
all  the  results,  divide  their  sum  by  2,  and  the  quotient  will 
be  the  area.     If  any  portion  of  the  boundary  is  curved, 
straight  lines  should  be  drawn  in  such  manner  as  to  enclose 
the  same  area  as  the  curve. 


EXAMPLES  FOR  THE  PUPIL.* 

1.  An  ox  measured  6ft.  Sin.  in  girth,  and  the  length  of 
his  back  was  5ft.  4in.  Determine  his  weight  by  each  of  the 
rules. 

2-5.  There  are  four  heaps  of  grain;  one  in  an  open 
field,  one  against  the  side  of  a  granary,  one  against  an 
inner  corner,  and  one  against  an  outer  corner  of  a  barn. 
The  depth  of  each  heap  is  4ft.  8in.,  and  the  slant  height 
6ft.  Tin.  Required  the  contents  in  bushels. 

6.  A  bin  8ft.  6in.  long,  3ft.  4in.  wide,  and  4ft.  9in.  deep, 
is  filled  with  ears  of  corn.     How  many  bushels  are  there, 
and  how  many  will  there  be  when  shelled  ? 

7.  How  much  hay  in  the  bay  of  a  barn,  the  bay  being 
18ft.  square,  and  the  hay  10ft.  deep  ? 

a  Colman,  Loudon,  G.  B.  Emerson,  Johnson,  Brit.  Huab.,  and  pri 
vate  information. 


i!4  THE   FARM.  [ART.  V. 

8.  How  many  furrows  9  inches  in  width,  will  make  an 
acre  in  a  field  that  is  6  rods  wide  ?   In  a  field  10  rods  wide  ? 
8  rods  wide  ?     40  rods  wide  ? 

9.  The  amount  of  wheat  annually  sown  in  France,  on 
10863959   acres,  is  estimated  at  32491978  bushels.     If, 
under  improved  cultivation,  £  of  this  amount  could  be  saved, 
and  the  crops  could  be  increased  5  bushels  per  acre,  what 
would  be  the  total  amount  gained  ? 

10.  If,  by  expending  $50  per  acre  in  the  cultivation  of 
potatoes,  one  farmer  can  secure  a  crop  of  300  bushels,  worth 
25  cents  a  bushel,  and  another  by  expending  $43.60  on  an 
acre  of  corn,  obtains  a  crop  worth  $67,  how  much  per  cent, 
does  the  former  gain,  by  cultivating  potatoes  instead  of 
corn  ?* 

11.  How  much  net  weight  must  a  pig  gain  per  day,  to 
pay  the  expense  of  keeping,  when  corn  sells  at  $1.00  per 
bushel,  and  pork  at  10  cents  per  lb.,  supposing  him  to  con 
sume  3qt.  of  corn  and  2cts.  worth  of  other  food  per  day  ? 

12.  If  3  lb.  live  weight  are  equivalent  to  2  lb.  net  weight, 
what  would  be  the  weekly  profit  of  a  pig  that  gains  2  lb. 
live  weight  per  day,  the  expense  of  keeping  and  the  price 
of  pork  being  the  same  as  in  the  last  example  ? 

13.  Produce  of  3J  acres  in  1841 :  21£bu.  wheat,  at  8s. ; 
44bu.  oats,  at  2s.  9d. ;  80bu.  potatoes,  at  Is.  ]  fodder  for  the 
support  of  2  calves,  at  £2  15s. ;  423 \  lb.  butter,  at  Is. ;  milk 
sold  or  given  to  the  pigs,  £10.     Kequired  the  value  of  the 
produce  per  acre,  at  $4.84  per  £. 

14.  It  is  estimated  that  in  Great  Britain  and  Ireland, 
7085370  acres  are  annually  sown  with  wheat,  at  the  rate  of 
2£bu.  per   acre.     How  many  quarters   of  8   bushels  are 
required  for  seed?     If  3£pk.  per  acre  would  be  sufficient, 

*  Although  the  crop  of  potatoes  may  be  more  profitable  for  a  single 
year  than  corn,  yet  a  rotation  of  crops  is  necessary  to  avoid  exhausting 
the  land. 


§22.]  MISCELLANEOUS   EXAMPLES.  115 

how  much  is  annually  wasted,  and  what  is  its  value,  at 
$1.12 £  per  bushel? 

15.  Find,  from  the  following  estimate  of  the  expense  of 
an  acre  of  corn,  the  balance  in  favor  of  the  crop. 

Expenses. — Ploughing,  $2.50;  harrowing,  $2.50;  holeing, 
50cts;  6bu.  leeched  ashes,  lOcts;  Ibu.  plaster,  65cts.;  lOqt. 
seed,  @  lOcts. ;  putting  on  ashes  and  plaster,  and  planting, 
$1.20;  harrowing,  30cts. ;  weeding,  $1.50;  cultivating, 
45cts. ;  second  and  third  hoeings,  $2.30;  gathering  and  husk 
ing,  $5.00;  gathering  stalks,  $1.50. 

Proceeds. — Crop. — 50bu.  corn  @  $1.00;  corn  fodder, 
equal  to  1  ton  of  hay,  $10. 

16.  How  many  hop-vines,  and  how  many  poles,  will  be 
required  to  an  acre,  allowing  two  poles  to  each  hill,  and  two 
vines  to  each  pole,  the  hills  being  eight  feet  apart; — and  what 
would  be  the  expense  of  the  poles  at  $3J  per  hundred?* 

Partial  Ans.  2720  vines. 

17.  Required  the  net  returns  on  an  acre  for  a  six  years' 
rotation  of  crops,  raising  corn  for  two  years,  wheat  the  third 
year,  and  hay  for  three  years ;  the  balance  in  favor  of  corn 
each  year  being  $17.75,  the  balance  in  favor  of  wheat,  $21, 
hay,  fourth  and  fifth  years,  each  $17.50,  sixth  year,  $11.50 ; — 
expenses  to  be  deducted, — grass-seed,  $1.87^,  and  six  years' 
compound  interest  on  land,  valued  at  $100  per  acre. 

18.  A  yoke  of  oxen  weighed  December  15,  4220  Ib. ; 
January  15,  4410  Ib. ;   March  7,  4730  Ib.     Required  the 
total  gain  from  December  15,  to  March  7,  and  the  average 
gain  per  day  from  December  15  to  January  15,  from  Janu 
ary  15  to  March  7,  and  from  December  15  to  March  7. 

19.  If  each  ox  in  the  preceding  example  consumed  daily 
14  Ib.  of  hay,  £bu.  of  potatoes,  and  8qt.  of  Indian  meal, 

a  To  determine  the  number  of  hills  or  plants  in  a  field,  divide  the 
area  of  the  field  by  the  area  occupied  by  each  hill  or  plant. 


116  THE   FARM.  [ART.  V. 

what  was  the  cost  of  the  gain  per  lb.,  supposing  hay  to  be 
worth  $10  per  ton,  potatoes,  20cts.  per  bushel,  and  Indian 
meal  GOcts.  per  bushel  ? 

20.  If  an  acre  of  land  yields  1£  tons  of  hay,  or  15  tons 
of  carrots  or  Swedish  turnips,   or  60  bushels  of  Indian 
corn ;  and  if  a  working  horse  would  consume  3cwt.  of  hay, 
or  3cwt.  of  carrots,  or  Ibu.  of  Indian  corn,  per  week,  how 
many  acres  of  each  crop  would  be  required  to  support  a 
horse  for  the  year  ? 

21.  Bought  December  1,  a  pair  of  oxen  for  $65,  and  sold 
the  same  February  26  at  $5  per  100  lb.,  net  weight,  their 
net  weight  being  1846  lb.     Required  the  amount  lost  on  the 
sale,  the  following  being  the  expenses  of  keeping : — 73bu. 
turnips  at  lOcts. ;  36 Ibu.  Indian  meal  at  60cts. ;  65ibu. 
potatoes  at  25cts.,  and  25  lb.  hay  per  day  at  25cts.  per  100  lb. ; 
interest  on  cost,  at  6  per  cent.,  94£cts.      Ans.  $24.44£. 

22.  What  is  the  value  of  pasture  land  per  acre,  50  acres 
of  which  will  pasture  8  cows  at  2 Sets,  per  week,  4  oxen  at 
50cts.,  and  75  sheep  at  3cts.  per  week  for  twenty  weeks  in 
the  year,  the  pasturage  being  estimated  at  6  per  cent,  of  the 
value  of  the  land  ? 

23.  Required  the  yearly  expense  of  keeping  a  sheep, 
allowing  155  days  for  the  time  of  foddering  in  the  barn, 
and  30  weeks  pasturage  at  3cts.  a  week,  the  food  consumed 
while  in  the  barn  being  2  lb.  of  hay  per  day  at  $8  per  ton 
of  2000  lb.,  and  Ubu.  of  rutabaga  at  lOcts.  a  bushel. 

24.  Required  the  amount  of  profit  in  the  following  experi 
ment  in  stall-feeding  sheep.     Bought  118  at  $2.50,  2  at  $3, 
and  60  at  $3^;   commissions  for  purchase  and  driving, 
25cts.  each ;  interest  and  risk  estimated  at  $9.27 ;  produce 
consumed,  519bu.  turnips  at  Sets.,  151bu.  corn  at  75cts., 
2  lb.  of  hay  each  per  day,  at  $8  per  ton  of  2000  lb.     The 
sheep  were  all  put  up  on  December  1st,  and  125  were  sold 
February  llth  at  $5  each,  and  the  others,  February  18th, 
at  $5.25  each. 


§22.] 


MISCELLANEOUS   EXAMPLES. 


117 


25.  Estimating  the  expense  of  cultivating  Indian  corn  at 
$25  per  acre,  wheat  at  $10,  oats  at  $5,  rye  at  $8,  hay  at  $2  ; 
and  the  harvest  per  acre  of  corn,  40bu.  at  65cts.,  and  fodder 
worth  $10;  wheat,  14bu.   at  $1.50,  and  straw  worth  $3; 
oats,  56bu.  at  42cts.,  and  straw  worth  $2.40;  rye,  12bu.  at 
$1,  and  straw  worth  $2.40 ;  and  hay,  2  tons  per  acre,  at  $10  ; 
what  would  be  the  profit  on  a  farm  containing  7J  acres  of 
corn,  5i  acres  of  wheat,  5  acres  of  oats,  5  acres  of  rye,  and 
21  acres  of  meadow  ? 

26.  When  hay  sells  at  $16  per  ton  of  2240  lb.,  and  corn 
stalks  at  let.  per  bundle,  how  much  can  be  saved  per  week 
with  6  cows,  by  substituting  corn-stalks  for  hay,  if  the 
average  consumption  of  each  cow  is  5  bundles  of  stalks,  or 
25  lb.  of  hay  per  day  ? 

27.  Two  hogs  were  fed  from  April  30th  to  May  20th, 
exclusively  upon  Indian  hasty-pudding.     The  pudding  used 
during  the  interval,  took  4Jbu.  of  meal,  at  78cts.  a  bushel, 
and  the  weight  gained  by  the  hogs  was  105  lb.     Allowing 
f  of  the  gain  to  be  net  weight,  what  should  be  the  price  of 
the  pork  per  lb.  to  gain  20  per  cent,  on  the  cost  ? 

28.  From  the  following  account  of  sales,  on  a  farm  of  25 
acres,  for  ten  years,  find  the  average  amount  received  annu 
ally  for  each  item. 


Date. 

Vegeta's. 

Fruit. 

Vinegar. 

Meat. 

Hay. 

Stock. 

Milk. 

Corn  and 
Barley. 

Total. 

1811 

$13'2.06 

$126.76 

35183.94 

$14  38'  .. 

1812 

181  .91 

145.48 

12.83 

81.  92  $130.  18 

$32.00 

$17.13 

$32.05 

1813 

112.93 

68.07 

93.47 

69.67 

202.68 

30.00 

44.97 

.75 

1814 

180.38 

63.71 

254.92 

41.21 

253.08 

112.00 

15.00 

103.50 

1815 

162.38 

151.56 

206.60 

61.37 

506.69 

16.99 

1816 

165.36 

169.73 

187.17 

60.69 

399.69 

37.00 

12.71 

128.50 

1817 

132.49 

240.37 

295.31 

48.24 

329.93 



24.75 

....,—  . 

1818 

84.31 

116.71 

246,30 

77.99 

162.21 

30.02 

98.47 

1819 

103.8,5 

280.68 

131.95 

25.84 

185.00 



94.04 



1820 

111.6-2 

248.88 

191.83 

87.21 

207.43 

128.27 

Tot. 

29.  What  was  the  average  net  annual  income  of  the  above 
farm,  estimating  the  amount  of  farm  produce  consumed  by 
the  family,  and  not  included  in  the  above  account,  at  $454, 
and  the  expenses  at  $638.30  per  annum  ? 


118  THE   FARM.  [ART.  V. 

30.  What  is  the  annual  gain  on  a  five-acre  wood   lot, 
which  cost  $50,  the  yearly  growth  being  1  cord  of  wood  per 
acre,  the  market  price  of  wood  averaging  $3.12  J  per  cord, 
and  the  expense  of  cutting  and  hauling  62 Jets,  per  cord? 

31.  Required  the  profit  on  an  acre  of  hops,  the  yield 
being  700  lb.,  worth  15  Jets,  a  lb.,  and  the  expenses  of  cul 
tivation  as  follows :  Renewal  and  setting  of  poles,  $12  ;  plant 
ing,  $1;  tying  up,  $1;  hoeing  3  times,  at  $1.50;  4  loads 
of  manure,  at  $1.12J;  picking,  let.  a  lb. ;  a  man  to  tend 
the  pickers,  $7;  board  of  the  laborers,  $1.50;  kiln-drying 
and  packing,  $1  per  100  lb. ;  bale,  45cts. 

32.  If  a  horse  consumes  the  produce  of  6  acres  of  land 
before  he  is  fit  for  work,  and  can  afterwards  be  kept  con 
stantly  employed  for  12  years,  by  annually  consuming  the 
produce  of  4  acres, — and  a  pair  of  oxen  consume  10  acres' 
produce  before  they  are  fit  to  work,  and  can  be  employed  only 
3  years  by  consuming  the  produce  of  2  J  acres  per  annum  each, 
the  produce  of  how  many  acres  will  represent  the  difference 
between  the  expense  of  keeping  a  horse  and  a  yoke  of  oxen 
for  12  years  ?  Ans.  46  acres. 

33.  Estimating  the  value  of  a  horse  after  12  years'  farm 
service,  at  $15,  and  supposing  that  every  ox  can  be  fattened 
after  3  years'  service  by  the  produce  of  1 J  acres,  so  as  to  be 
worth  $65,  what  should  be  the  value  of  1  acre's  produce,  to 
make  the  expense  of  keeping,  the  same  for  a  horse  as  for  a 
yoke  of  oxen?  Ans.  $8.71  nearly. 

N.  B.  Add  to  4(5  acres,  the  number  of  acres'  produce  necessary 
to  fatten  the  oxen,  and  you  will  obtain  the  number  of  acres' 
produce  equivalent  to  the  difference  of  value  between  the  horse 
and  the  oxen. 

34.  Find  the  expense  of  keeping  8  oxen  one  year,  esti 
mating  their  total  consumption  at  1  ton  of  hay  per  week 
from  January  1  to  May  15,  and  from  October  30th  to 
January  1,  the  hay  being  worth  $10.75  per  ton,  pasturage 


§22.]  MISCELLANEOUS   EXAMPLES.  119 

from  May  15  to  October  30,  at  62 Jets,  per  week  each, 
repairs  of  yokes  and  bows  $3.75,  wear  of  ploughs,  chains, 
&c.  $15. 

35.  What  should  be  the  price  of  charcoal  per  bushel,  to 
make  25  per  cent,  on  the  following  estimate  of  the  cost  of 
burning  a  kiln  of  30  cords,  the  wages  of  labor  being  $1.25 
per  day  ?     Cost  of  standing  wood,  $1.75  per  cord ;  2£  cords 
make  100  bushels  of  coal  j  1  man  can  chop  2  cords  for  the 
kiln  in  a  day  j  collecting,  drawing  together,  covering,  and 
burning,  require  20  days'  work  of  1  man ;  expense  of  haul 
ing  the  whole  to  market,  and  selling,  $8.75. 

Ans.   10 A  |  cents. 

36.  The  whole  amount  of  hay  purchased  and  used  at  a 
stage  stable,  from  April  1  to  October  1,  1816,  was  32T. 
4cwt.  10  Ib.  at  an  average  cost  of  $25  per  ton.     At  the 
same  stable,  from  October  1,  1816,  to  April  1,  1817,  there 
was  consumed  by  the  same  number  of  horses,  16T.  locwt. 
3qr.  10  Ib.  of  straw,  at  $9.75  per  ton,  and  9T.  14cwt.  Iqr. 
of  hay  at  $25  per  ton.    A  straw-cutter  was  employed  during 
four  months  of  the  latter  period.     Required  the  amount 
of  money  saved  by  its  use. 

37.  In  an  experiment  on  transplanting  the  layers  of  a 
single  stalk  of  wheat,  made  by  Mr.  Miller,  of  Cambridge, 
Eng.,  one  grain   of  wheat  produced  3f  pecks,  weighing 
47  Ib.  7oz.     Supposing  a  cubic  inch  to  contain  200  grains, 
how  many  grains  were  there  in  the  whole  ?     How  many  in 
a  pound  ? 

38.  If  6  men,  with  cradles,  can  cut  as  much  grain  in  a 
day  as  12  men  with  common  sickles,  and  if  the  6  cradlers 
require  3  men  to  bind,  and  3  boys  to  assist,  while  the  12 
reapers  require  only  2  men  to  bind,  how  much  can  be  saved 
per  day  by  cradling,  allowing  $1.25  for  each  man,  and  50 
cents  for  each  boy  ? 

39.  If  105  gallons  of  milk  yield  36  Ib.  of  butter,  worth 


120  THE   FARM.  [ART.  V. 

18  fcts.  a  pound,  and  GOlb.  skim-milk  cheese,  worth  Gets,  a 
pound,  and  if  140  gallons  of  milk  of  the  same  quality, 
with  the  same  amount  of  labor,  yield  141  lb.  of  cheese, 
worth  9cts.  a  lb.,  and  whey  worth  $lf,  which  is  the  more 
profitable,  the  manufacture  of  butter  or  cheese  ? 

40.  It  is  supposed  that  a  pair  of  tame  pigeons  consume  a 
pint  of  grain  per  day,  for  280  days  in  the  year.     As  it  has 
been  estimated  that  there  are  1250000  pairs  in  Great  Bri 
tain,  how  much  would  they  consume  per  annum  at  this  rate, 
and  what  would  be  its  value,  at  an  average  of  50  cents  a 
bushel  ? 

41.  The  cost  of  improving  a  12-acre  field  was  as  follows  : 
Blasting   large    stones,    $44.25;    trenching,    $20;   drains, 
$3.87i;  lime,  64bu.  per  acre,  at  40cts.  per  bu. ;  ditch  for 
intercepting  hill  water,  837.62J.     Required  the  total  cost, 
and  the  average  per  acre. 

42.  What  should  be  the  market  price  of  milk  per  quart, 
when  butter  is  worth  25cts.  a  pound,  and  skim-milk  cheese 
is  worth  Sets,  a  pound, — supposing  that  100  gallons  of  milk 
will  make  34  lb.  of  butter,  and  74  lb.  of  skim-milk  cheese  ? 

43.  The  late  Duke  of  Athol  planted  6500  Scotch  acres* 
of  mountain  ground  with  the  larch,  which,  in  72  years  from 
the  time  of  planting,  will  be  a  forest  of  timber  fit  for  build 
ing  ships  of  the  largest  class.     Supposing  the  plantation  by 
that  time  to  be  thinned  out  to  400  trees  per  acre,  and  each 
tree  to  contain  one  load  of  50  c.  ft.,  what  will  be  the  value 
of  the  wood  per  acre,  and  the  value  of  the  whole  plantation 
at  25cts.  per  c.  ft.,  which  is  about  £  its  present  value  ? 

44.  A  hectare  in  the  Isle  of  Bourbon,  produces  76000 
kilogrammes  of  cane,  which  give  9200   kilogrammes  of 
sugar,  at  an  expense  of  2500  francs  for  labor.     A  hectare 
of  beet-root  produces  40000  kilogrammes  of  roots,  which 


a  A  Scotch  acre  =  about  U  English  acres. 


§22.]  MISCELLANEOUS   EXAMPLES.  121 

will  yield  2400  kilogrammes  of  sugar,  at  an  expense  of  354 
francs.  What  per  centage  of  sugar  is  yielded  by  the  cane,  and 
by  the  beet-root  ?  What  is  the  cost  of  each  kind  of  sugar  per 
lb.,  estimating  merely  the  labor  bestowed  on  each  crop  ? 

45.  Two  men  mow  a  square  meadow,  but  one  being  a 
faster  mower  than  the  other,  agrees  to  take  the  outside 
swath,  and  cut  off  all  the  corners.     What  part  of  the  whole 
will  each  mow,  there  being  twelve  swaths  in  each  side  of 
the  field?  Ans.  ||,  41. 

The  following  rule  will  furnish  the  answer  to  all  questions  of 
this  kind.  Let  the  pupil  endeavor  to  prove  its  accuracy. 

Square  the  number  of  swaths  in  the  side  of  the  field  for  a 
denominator.  Then  if  the  number  of  swaths  is  odd,  multiply  it 
by  2,  and  diminish  the  product  by  1 ;  or,  if  it  is  even,  multiply  it 
by  2,  and  diminish  the  product  by  4,  for  a  numerator.  The  frac 
tion  thus  obtained,  will  show  what  part  of  the  field  the  outer  man 
will  mow  more  than  the  inner  one. 

46.  The  amount  of  hay  necessary  to  sustain   oxen,  is 
about  .02  of  their  weight,  daily.     When  fattening,  they 
require  about  .04  of  their  weight  per  day.     At  $10.50  per 
ton  of  2000  lb.,  what  would  be  the  cost  per  week  of  the  extra 
hay  consumed  in  fattening  two  oxen,  one  of  which  weighs 
834  lb.,  and  the  other  917  lb.  ? 

47.  In  England  and  Wales,  it  has  been  estimated  that 
there  are  3252000  acres  of  wheat,  1250000 A.  of  barley  and 
rye,  3200000  A.  of  oats,  beans  and  peas,  1200000  A.  of  clover 
and  artificial  grasses,  1200000A.  of  field-roots,  2100000A. 
fallow,  48000A.  of  hops,  17300000A.  of  meadow  and  past 
ure,  1200000A.  of  hedgerows,  copses,  woods,  and  wastes, 
and  that  the  annual  agricultural  income  is  £216817624 
At  $4.84  per  pound,  what  is  the  average  income  per  acre  ? 

48.  What  would  be  the  cost  of  seed  in  the  last  example, 
supposing  that  there  are  500000  acres  of  barley,  and  that  the 
following  is  the  average  quantity  used,  per  acre  ?     Wheat, 


122  THE   FARM.  [ART.  V. 

5pk.  @  $1.10  per  bu.;  barley,  3bu.  @  §0.50;  rye,  Ibu. 
@  $0.80 ;  oats,  beans,  and  peas,  3bu.  @  $0.95 ;  grasses, 
opk.  @  $2.00  per  bu. ;  field-roots,  13bu.  @  $0.45. 

49.  A  farmer  wishing  to  determine  the  area  of  his  field, 
finds  that  it  may  be  divided  into  four  triangles,  the  dimen 
sions  of  which  are  as  follows  :    1st.  Base  240  paces,  altitude 
87  paces.     2d.    Base  213  paces,  altitude  95  paces.     3d. 
Base  107  paces,  altitude  28  paces.     4th.  Base  92  paces, 
altitude  79  paces.     Required  the  entire  area. 

Ans.  5 A.  1R.  9  Jr.  nearly. 

50.  I  have  a  barrel  20  inches  in  diameter  at  the  middle 
point,  16  inches  at  the  head,  and  27  inches  long.     Required 
its  contents  in  ale,  wine,  and  dry  measure. 

Ans.  Slgal.  nearly,  wine  meas. 
25gal.  liqt.  beer  meas. 
3bu.  Ipk.  2qt.  dry  meas. 

51.*  A  heap  of  grain  piled  against  the  outer  corner  of  a 
barn,  contains  63  bushels,  and  its  depth  is  5ft.  Required 
the  slant  height.  Ans.  6ft.  Sin. 

52.  A  granary  holds  1310.4  bushels.     The  dimensions 
of  the  floor  are  15ft.  9in.  by  12ft.     Required  the  average 
depth  of  the  grain.  Ans.  8ft.  Sin. 

53.  A  stack  of  hay  with  a  circular  base,  has  settled  so 
much  that  250  c.  ft.  are  estimated  as  equivalent  to  1  ton, 
and  according  to  this  estimate,  it  is  found  to  contain  1.44 
tons.     What  is  the  height  of  the  stack,  the  circumference 
of  the  base  being  30ft.  ?  Ans.  16ft. 

54.  The  area  of  an  irregular  field  is  2A.  3R.,  and  the 
average  length  is  121  paces.    What  is  the  average  breadth  ? 

Ans.  110  paces. 

*  The  remaining  examples  in  this  section  are  introduced  to  test  the 
skill  of  the  pupil  in  reversing  the  rules. 


§23.]  MISCELLANEOUS   EXAMPLES.  123 


VI.    THE  GARDEN. 

EXAMPLES  FOR  THE  PUPIL/ 

1—5.  How  many  plants  would  be  required  for  an  acre,  if 
they  were  placed  l£ft.  apart?  If  placed  2ft.  apart  ?  3ft. 
apart?  6ft.  apart?  16Jft.  apart? 

Am.  to  the  first,  19360. 

6-10.  How  long  a  strip  of  land  will  be  required  to  con 
tain  an  acre,  if  the  width  is  100ft.  ?  If  the  width  is  3f  rods  ? 
97.6ft.?  18j3rft.?  2r.  5Jft.? 

11.  Allowing  Ipt.  of  early  peas  to  a  row  20yd.  long,  Ipt. 
of  marrowfat  peas  to  a  row  32yd.  long,  Ipt.  of  string  beans 
to  a  row  27yd.  long,  Ipt.  of  runners  to  a  row  36yd.  long,  and 
Ipt.  of  dwarf  kidney  beans  to  a  row  26yd.  long,  how  much 
seed  would  be  required  to  plant  4  rows  of  early  peas,  6  rows 
of  marrowfats,  3  rows  of  string  beans,  5  rows  of  runners, 
and  8  rows  of  dwarf  kidney  beans,  each  row  measuring  28 
yards  ? 

12.  A  quarter  acre  of  land  cost  $25.    Expended  for  labor 
in  improving  it,  $157 ;   for  seed  potatoes,  $15  ;  rye  and 
grass  seed,  $1.17;  6  cords  of  manure,  at  $5  per  cord;  2 
casks  of  lime  at  $1 ;  cost  of  other  seeds  and  gathering  in 
the  crop,  $17.50.     The  products  of  the  first  year  were  327 
bushels  of  potatoes,  at  60cts. ;  5bu.  rye,  at  $1.25;  8-Jbu. 
corn,  at  $1 ;  lOObu.  rutabaga,   at  30cts. ;  hay,  $12 ;  600 
cabbages,  at  50cts.  a  dozen  ;    2000  Ib.  squashes,  at  let. ; 
fuel  taken  off,  $25.b    Estimating  the  value  of  the  land  as 
doubled,  what  was  the  first  year's  balance  in  favor  of  the 
improvement?  Ans.  $125.28. 

13-15.  How  many  times  must  a  spade  be  thrust  into  the 
ground  in  digging  a  square  ro'd,  supposing  the  surface  removed 

*  Buist,  Colman,  Loudon,  Johnson.  b  Colman. 


THE   GARDEN.  [ART.  VI, 

at  each  time  to  measure  7  by  8  inches?  The  spade 
weighing  81b.,  and  a  spadeful  of  earth  17  lb.,  how  many 
pounds  must  the  gardener  lift  in  a  day,  10  sq.  rods  being  a 
day's  work?  How  many  tons  must  he  lift  in  spading  an 


acre .' 


16.  What  amount  is  annually  gained  by  expending  §16 
per  acre  to  drain  a  garden  of  If  acres,  money  being  worth 
6  per  cent,  per  annum,  and  the  land  renting  before  it  was 
drained  for  83  per  acre,  but  after  it  was  drained,  for  88 
per  acre  ? 

17.  What  distance  would  a  man  walk  in  ploughing  a  gar 
den  containing  f  of  an  acre,  if  each  furrow  was  9°  inches 
wide,  adding  1  rod  to  every  18  for  the  ground  travelled  over 
in  turning  ? 

18.  What  must  be  the  length  of  a  strawberry  bed,  to 
contain  ^  of  an  acre,  the  width  being  33  feet  ? 

19.  Required  the  cost  of  digging  a  trench  200ft.  long,  3ft. 
broad,  and  3ft.  deep,  a  laborer  being  able  to  remove  1  cubic 
yard  per  hour,— allowing  for  wages  81.25  per  day  of  10 
hours. 

20.  Money  being  worth  6  per  cent,  simple  interest,  what 
is  the  present  value  of  an  orchard  containing  150  trees, 
which  will  be  all  in  full  bearing  in  5  years,  and  will  then  be 
worth  820  apiece,  the  land  which  they  occupy  being  now 
worth  81450  ?  Am.  83757.69. 

21.  In  a  garden  of  1£  acres  the  following  articles  were 
raised  in  the  year  1836  :— 3500bu.  onions,  at  5cts. ;  45bbl. 
beets,  at  81.50;   14bu.  parsnips,  at  75cts. ;   2bu.  beans,  at 
82;  20bu.  potatoes,  at  8i;   8100  worth  of  cabbages;  and 
vegetables  for  the   family,  estimated  at  8100.     Allowing 
one-half  of  the  proceeds  for  the  'expense  of  cultivation,  and 
estimating  the   value  of  the  land  at  8450  an  acre,  what 
profit  was  made  on  the  whole  ? 


§23.]  MISCELLANEOUS   EXAMPLES.  125 

22.  When  ordinary  pears  are  sold  at  50  cents  a  bushel, 
and  choice  varieties  bring  3  cents  apiece,  how  much  more 
profitable  is  a  tree  grafted  with  one  of  the  best  kinds,  than 
a  common  pear-tree,  the  yield  of  each  being  eight  bushels, 
and  the  choice  pears  averaging  45  to  a  peck  ? 

23.  In  a  garden  225ft.  long,  and  144ft.  broad,  is  a  gravel 
walk  6ft.  wide,  extending  around  the  whole,  at  a  uniform 
distance  of  8ft.  from  the  boundary  line,  intersected  by  three 
walks  each  4ft.  wide,  parallel  to  the  shortest  sides.     What 
amount  of  land  is  taken  up  by  the  walks  ? 

Ans.  650x6  +  348x4  =  5292  sq.  ft. 

24.  Throughout  the  whole  extent  of  the  above  walks  is  a 
trench  lOin.  wide  and  Sin.  deep,  filled  with  broken  stone,  and 
covered  with  a  layer  of  broken  bricks  and  other  rubbish, 
3in.  thick  over  the  whole  width  of  the  walks,  the  top-dress 
ing  of  gravel  being  of  the  average  depth  of  4  inches.     How 
many  loads  of  27c.  ft.  each,  were  required  of  the  broken 
stone?     How  many  of  rubbish?     How  many  of  gravel? 
What  was  the  cost  of  the  whole  at  50cts.  for  each  load  ? 

Ans.  20£f  §  loads  of  stone,  49  of  rubbish, 
65 J  of  gravel;  cost,  $ 

25.  What  would  be  the  expense  of  making  a  hedge  of 
roses  468ft.  long,  setting  the  bushes  with  a  distance  of  6£ 
inches  between  the  centres,  the  cost  of  the  bushes  being 
28cts.  apiece,  and  the  expense  of  setting  $2.25  per  hundred  ? 

26.  I  wish  to  make  a  hot-bed,  composed  of  3  parts  manure 
and  1  part  oak  leaves.     The  frame  is  12ft.  long  and  6ft. 
wide.    The  bed  is  to  be  3  J  ft.  deep,  and  is  to  project  8  inches 
beyond  the  frame,  on  every  side.     How  many  cubic  yards 
of  manure  will  be  required,  and  what  will  it  cost  at  $1.75 
per  yard  ? 

27.  Two  banks,  each  104ft.  long,  and  51ft.  high,  and  a 
grass  plat  162ft.  long  and  157ft.  wide,  are  to  be  covered  with 
turf.     Estimating  the  average  thickness  of  the  turf  at  3in., 


126  THE   GARDEN.  [ART.  VI. 

what  would  be  the  cost  of  the  whole  at  $1.25  per  c.  yd? 
At  IJcts.  per  sq.  ft.  ? 

28.  An  English  market-gardener  received  for  the  produce 
of  a  single  acre  in  one  year,  for  radishes,  £10;  cauliflowers, 
£60 ;  cabbages,  £30 ;  celery,  £90 ;  endive,  £30.     Allowing 
£3  10s.  for  rent,  and  £90  for  expenses  of  cultivation  and 
marketing,  what  was  the  profit  per  sq.  rod,  in  Federal  Money, 
estimating  the  value  of  the  sovereign  at  $4.84  ? 

29.  Frederick  Tudor,  Esq.,  of  Nahant,  Mass.,  in  the  year 
1841,  raised  42284  Ib.  of  sugar-beets  on  93  rods  of  land. 
Allowing  56  Ib.  per  bushel,  how  many  bushels  could  be 
raised  per  acre  at  this  rate,  and  what  would  be  the  value  of 
the  whole  at  50cts.  per  bushel  ? 

30.  If  2  men  digging  can  keep  1  man  employed  in  wheel 
ing  to  the  distance  of  20  yards,  and  if  each  digger  removes 
15  c.  yd.  per  day,  what  will  it  cost  to  remove  3340  c.  yd.  to 
the  distance  of  160  yards,  the  wages  of  each  laborer  being 
75cts.  per  day  ?  Ans.  $835. 

N.  B.  First  find  the  number  of  men  required  to  wheel  the 
dirt  away,  and  add  the  two  diggers,  and  you  will  obtain  the  whole 
number  employed. 

31.  What  number  of  bulbs  will  be  required  for  a  crocus 
bed,  the  bed  being  23in.  wide,  and  the  rows  6  inches  apart, 
allowing  150  bulbs  to  each  row;  and  what  will  they  cost,  $  of 
the  whole  number  being  bought  at  75cts.  per  100,  £  at 
$1.37i,  and  the  rest  at  $1.62£  ? 

32.  When  the  double  hyacinths  were  first  brought  into 
notice,  some  of  the  roots  were  sold  at  2000  guilders  apiece. 
Equally  fine  varieties  can  now  be  bought  for  $4  a  dozen. 
At  40cts.  per  guilder,  how  large  a  bed  could  now  be  stocked 
with  the  choicest  varieties,  at  the  original  cost  of  a  single 
root,  the  roots  being  placed  8  inches  apart,  and  allowing  6in. 
for  border  ? 

33.  What  will  be  the  cost,  at  $12.75  per  100,  of  stocking 
a  tulip  bed  20ft.  long  and  4ft.  wide,  the  bulbs  being  planted 


§23.]  MISCELLANEOUS   EXAMPLES.  127 

in  rows,  allowing  6in.  between  the  plants,  and  Tin.  between 
the  rows?  Ans.  $35.70. 

34.  In  order  to  introduce  enough  fresh  air  to  fill  a  hot 
house  twice  in  24  hours,  how  many  feet  of  air  heated  to 
the  proper  temperature,  must  be  introduced  hourly  into  a 
house  40ft.  long,  and  16ft.  wide,  the  height  in  front  being 
6ft.,  and  at  the  back  18ft.  ?  Ans.  320  c.  ft. 

35.  What  would  be  the  daily  expense  of  raising  the  tem 
perature  of  the  hot-house  in  the  last  example  15°,  if  2sA(J3  lb. 
of  coal  will  raise  the  heat  of  a  c.  ft.  of  air  1°,  coal  being 
worth  $6.50  per  ton  of  2000  Ib.  ? 

36.  A  mechanic  has  a  vacant  space  in  his  garden,  18ft. 
long  and  4ft.  wide,  which  he  wishes  to  occupy  as  an  asparagus 
bed.     He  trenches  the  whole  to  the  depth  of  2ft.,  and  fills 
the  trench  half  full  of  manure  before  returning  the  earth. 
He  afterwards  plants  the  whole  with  roots,  setting  the  roots 
9  inches  apart.     The   land  is  worth  12£cts.  a  square  foot, 
and  he  will  be  obliged  to  wait  3  years  before  the  bed  will  be 
in  a  condition  to  be  cut.     Estimating  the  labor  bestowed 
on  the  bed  at  4  hours,  at  lOcts.  per  hour,  the  cost  of 
manure  at  $1.25  per  solid  yard,  the  cost  of  plants  at  $1.00 
per  100,  and  interest  on  the  whole  at  6  per  cent,  per  annum, 
what  will  be  the  entire  cost  of  the  bed  when  it  comes  into 
bearing  ? 

37.  Allowing  loz.  of  onion,  carrot,  or  parsnip  seed   for 
sowing  15  sq.  yd.,  £oz.  of  cabbage  or  cauliflower  seed  for 
4  sq.  yd.,  $oz.  of  turnip  seed  for  11  sq.  yd.,  and  160  aspa 
ragus  plants  for  a  bed  5ft.  by  30,  what  quantity  of  each 
will  be  required  to  plant  a  garden  of  40  sq.  rods  with  onions, 
carrots,  parsnips,  cabbages,  cauliflowers,  turnips,  and  aspa 
ragus,  allowing  the  same  quantity  of  ground  to  each  ? 

38.  In  1637  a  collection  of  120  tulips  was  sold  in  Hol 
land  for  9000  guilders ;  in  England,  at  the  present  day, 
£50  is  frequently  given  for  a  single  bulb  no  finer  than  some 


128  THE    HOUSEHOLD.  [ART.  VII. 

of  the  varieties  which  can  be  purchased  for  less  than  a  dol 
lar.  Estimating  the  florin  at  40  cents,  and  the  sovereign 
at  §4.84,  how  many  of  the  Dutch  bulbs  would  amount  to 
the  same  as  15  of  the  English  bulbs,  at  the  above  rates  ? 

39.  Apples  should  stand  35  feet  apart,  pears  20ft.  apart, 
and  plums  18ft.  apart.  How  large  an  orchard  would  be 
required  for  180  apple-trees,  315  pear-trees,  and  350  plum- 
trees,  there  being  5  rows  of  each  sort,  allowing  28ft.  between 
the  apples  and  pears,  and  20ft.  between  the  pears  and 
plums?  Am.  11A.  3R.  31r.  12iyd. 

40-42.  How  many  hills  of  corn  in  a  rectangular  piece 
of  land  containing  a  quarter  of  an  acre,  if  the  hills  are  3ft. 
apart  one  way,  and  2ft.  9in.  the  other  ?  if  the  hills  are  2ft. 
Gin.  X  3ft  ?— 2ft,  x  2ft.  3in.  ? 


VII.    THE  HOUSEHOLD. 
94.     GENERAL  INFORMATION. 

1.  THE  grains  yield  nearly  the  following  quantities  of 
meal  and  bread  per  bushel.*    Wheat  weighing  60  Ib. — flour, 
48  Ib. ;  bread,  64  Ib.     Rye  weighing  54  Ib.— meal,  42  Ib. ; 
bread,  56  Ib.     Barley  weighing  48  Ib. — meal,  37 2  Ib. ;  bread, 
50  Ib.     Oats  weighing  40  Ib.— meal,  22  J  Ib. ;  bread,  30  Ib. 

2.  The  following  weights  and  measures  nearly  correspond.5 
Wheat  flour,  14oz.  =  Ipt,     Indian  meal,  18oz.  =  Ipt.     But 
ter,  15oz.  — Ipt.     Loaf  sugar,  broken,  llb.  =  lpt.     White 
sugar,    crushed,    17oz.  =  lpt.     Brown    sugar,    18oz.  — Ipt. 
Eggs,  10  =  1  Ib.     A  chaldron  of  soft  coal  =  58f  c.  ft.     Stone 
coal,  lbu.  =  70  Ib.,  or  42  c.  ft.  =  l  ton.     A  common  tumbler 
holds  ipt.  or  £  Ib.  of  water.     A  common  wine-glass  holds 

a  Brit.  Husbandry. 

3  Sci.  American,  Brit.  Husb.,  Enc.  Brit.,  and  private  information. 


§25.]          HOUSEHOLD  MENSURATION.          129 

£  gill  or  2oz.  of  water.  Currants,  14oz.=lpt.  Cherries, 
12oz.  =  lpt.  Honey,  23oz.  =  Ipt.  Lard,  tallow,  or  sperma 
ceti,  15oz.=  lpt.  Milk,  lib.  =  Ipt.  Qil,  15oz.  =  Ipt. 
Salaeratus,  dry,  23oz.=lpt. 

25.    HOUSEHOLD  MENSURATION. 

1.  To  find  the  contents  of  any  cylindrical  vessel,  in  pints. 
—  Measure  the  diameter  and  height  in  inches.     Then,  for 
wine  measure,  multiply  the  square  of  the  diameter  by  the 
height,  and  divide  the  product  by  37  ;  for  beer  measure, 
multiply  the  square  of  twice  the  diameter  by  twice  the 
height,  and  divide  the  product  by  359  ;  for  dry  measure, 
multiply  the  square  of  twice  the  diameter  by  the  height, 
and  divide  the  product  by  171. 

Let  D  represent  the  diameter,  H  the  height,  and  C  the 
contents  in  pints.  Then,  if  the  contents  are  taken  in  wine 
measure,  H  =  37C  -f-D2  ;  D=  V37CH-H.  For  beer  meas- 
.ure  substitute  441,  and  for  dry  measure  42  f,  in  the  place 
of  37,  in  each  formula. 

2.  To  find  the  contents  of  any  vessel  with  a  circular  base, 
and  tapering  sides*  —  Take   one  half  of  the  sum  of  the 
greater  and  less  diameters,  for  a  mean  diameter,  and  pro 
ceed  with  this  mean  diameter  and  the  height,  as  in  the  pre 
ceding  problem.     If  great  accuracy  is  required,  proceed  as 
in  Problem  1,  substituting  for  "the  square  of  the  diam 
eter,"  "  the  product  of  the  greater  and  less  diameters,  added 
to  one  third  of  the  square  of  their  difference."     In  beer 
and  dry  measure,  use  four  times  this  product,  in  the  place 
of  "  the  square  of  twice  the  diameter." 

The  height  and  mean  diameter,  may  be  found  as  in  Prob. 
1.  Let  G  be  the  greater  diameter,  L  the  less  diameter, 
and  M  the  mean  diameter.  Then 

-f  of  La-}L 


L  =  ^3M*-i  of  G3-J  G. 


*  Such  a  vessel  forms  &  frustum  of  a  cone. 


130  THE   HOUSEHOLD.  [ART.  VII. 

3.  To  find  the,  contents  of  a  bowl,  in  pin te.— Measure  the 
diameter  of  the  top  and  the  depth,  in  inches.     To  three 
times  the  square  of  half  the  top  diameter,  add  the  square 
of  the  depth ;  multiply  this  sum  by  the  depth,  and  divide 
the   product   by   55   for  wine   measure,  by  67   for   beer 
measure,  or  by  64  for  dry  measure. 

4.  To  find  the  contents  of  a  icell  or  cylindrical  cistern  in 
hogsheads. — Measure  the  dimensions  in  feet,  multiply  the 
square  of  the  diameter  by  the  depth,  and  divide  the  product 
by  11.     If  great  accuracy  is  required,  measure  the  dimen 
sions  in  inches }  multiply  the  square  of  the  diameter  by  the 
depth ;  multiply  the  product  by  54,  and  cut  off  six  figures 
from  the  right  hand. 

Let  D  be  the  diameter  in  inches,  d  the  depth,  and  C  the 
contents  in  hogsheads.  Then  D  =  V 1000000  C  -^  54  d ; 
d=  1000000  CH-  54  D2. 

5.  To  determine  the  height  to  which  any  cylindrical  vessel 
will  be  filled  by  a  gallon. — For  wine  measure,  divide  294 ; 
for  beer  measure,  divide  359 ;  for  dry  measure,  divide  342, 
by  the  square  of  the  diameter,  measured  in  inches. 

One  quart  would  evidently  fill  the  vessel  \  as  high; 
one  pint  would  fill  it  £  as  high }  and  one  gill  would  fill  it  ^ 
as  high.  Any  cylindrical  vessel  can  therefore  be  marked  by 
this  rule,  so  as  to  serve  the  purpose  of  a  set  of  measures. 

Let  N  be  the  number  of  the  required  measure  which  will 
be  equivalent  to  a  gallon,  D  the  diameter,  and  H  the 
height.  Then,  for  wine  measure  D=  \/294-r-  (N  x  H) ;  for 
beer  measure  D=  \/359-f-  (N  x  H) ;  for  dry  measure  D= 
V342-r(NxH.) 

6.  To  test  tic  accuracy  of  cylindrical  dry  measures, — 
Measure  the  diameter  in  inches,  and  divide  2738  by  the 
square  of.  the  diameter.     The  quotient  will  be  the  depth  for 
1  bushel.     The  depth  for   a  half  bushel,  will   be  £  the 


§26.]  MISCELLANEOUS  EXAMPLES.  131 

quotient ;  for  a  peck,  i  of  the  quotient ;  for  a  half  peck,  J 
of  the  quotient,  &c. 

36.    EXAMPLES  FOR  THE  PUPIL. 

1-3.  I  have  a  cylindrical  tin  dish,  6£in.  in  diameter,  and 

Gin.  deep.     Required  its  contents  in  wine,  in  heer,  and  in 

dry  measure.         Ans.  3qt.  3.4gi.  wine  meas. ;  2qt.  1.65pt. 

beer  meas. ;  3qt.  nearly,  dry  meas. 

4-6.  At  what  height  should  marks  be  placed  on  the  inside 
of  the  above  dish,  to  indicate  a  quart  of  each  measure  ? 

Ans.  If  in.  nearly  for  wine  meas. ;  2|in.  nearly 
for  beer  meas. ;  2in.  for  dry  meas. 

7-9.  Find  the  contents  in  each  measure,  of  a  pan,  the 
height  being  5  inches,  the  top  diameter  17in.,  and  the 
bottom  diameter  9in. 

Ans.  by  the  accurate  rule,  23.56pt.  wine  meas. ; 
19.42pt.  beer  meas.;  20.39pt.  dry  meas. 

10-12.  Find  the  contents  in  each  measure,  of  a  bowl,  the 
diameter  of  which  is  6in.,  and  the  depth  3in. 

Ans.  1.96pt.  wine  meas. ;  1.61pt.  beer  meas. ; 
1.69pt.  dry  meas. 

13.  A  cylindrical  cistern  is  5  feet  in  diameter,  and  6ft. 
deep.  How  many  hogsheads  does  it  hold  ? 

14—15.  A  cylindrical  cistern  is  7ft.  6in.  in  diameter,  and 
6ft.  9in.  deep.  Find  its  contents  by  each  rule. 

Ans.  by  Rule  1,  34TVehhd. ;  by  Rule  2,  35.43hhd. 

16-17.  The  house  to  which  the  above  cistern  belongs,  is 
40ft.  long,  and  34ft.  wide,  and  is  supplied  with  eave-troughs 
which  convey  all  the  water  that  falls  on  the  roof  to  the  cis 
tern.  To  what  depth  would  the  cistern  be  filled  by  a  single 
shower,  in  which  there  is  a  fall  of  fin.  of  rain,  allowing  6in. 
on  each  side  for  the  projection  of  the  eaves  ?  How  many 
inches  of  rain  would  fill  the  cistern  ? 

18.  A  man  has  the  following  cylindrical  measures :  One 


132  THE   HOUSEHOLD.  [AKT.  VII. 

designed  for  a  bushel,  the  diameter  of  which  is  18  J  inches; 
one  designed  for  a  half-bushel,  diameter  14 £in.  ;  one  de 
signed  for  a  peck,  diameter  12in. ;  one  designed  for  a  half- 
peck,  diameter  9iin. ;  and  one  designed  for  a  quarter-peck, 
diameter  7£in.  What  should  be  the  depth  of  each  ? 

Ans.  for  the  half-bushel,  6£in. 

19.  I  send  a  barrel  of  flour  to  a  baker,  and  he  agrees  to 
furnish  me  an  equal  weight  of  bread  in  return.     The  flour 
weighs  Icwt.  3qr.,  and  cost  me  $7.50;  the  barrel,  (which 
the  baker  keeps,)  is  worth  25cts.     How  much  do  I  pay  the 
baker  for  his  trouble  ?  Ans.  $2.12£. 

N.  B.  If  48  Ib.  of  flour  make  64  Ib.  of  bread,  how  many  Ib.  of 
flour  will  make  Icwt.  3qr.  of  bread  ?  And  what  will  be  the  value 
of  the  flour  that  is  left,  at  $7.50  per  bbl.  ? 

20.  A  cow  gave  350  gallons  of  milk  in  a  year,  and  con 
sumed  in  the  same  time  two  tons  of  hay,  at  $9.75;  12bu. 
of  Indian  meal,  at  GOcts. ;  50bu.  of  beets,  at  40cts. ;  and 
her  pasturage  cost  $8.75.     Estimating  the  value  of  her 
milk  at  5cts.  a  quart,  and  allowing  10  per  cent,  on  $25  for 
the  interest  of  her  worth,  and  risk  of  keeping,  and  $2.50  for 
labor  and  attendance,  what  amount  of  profit  did  she  yield  ? 

21.  Find  the  cost  of  100  Ib.  of  bread,  made  from  each 
of  the  following  grains,  the  grain  being  of  the  full  weight 
given  in  §  24,  and  the  cost  of  manufacture  in  each  instance, 
being  56cts. ;  wheat  at  $1.10  per  bushel;  rye,  at  90cts.  per 
bu. ;  barley,  at  75cts.  per  bu. ;  oats,  at  40cts.  per  bu. 

22.  How  much  crushed  sugar,  by  measure,  should  be  used 
in  preserving  6qt.  of  currants,  in  order  that  there  may  be 
equal  weights  of  sugar  and  fruit?* 

23.  How  would  you  measure  the  following  ingredients  ? — 
2  Ib.  flour,  2  Ib.  sugar,  1  Ib.  butter,  and  1  Ib.  eggs.a 

24.  A  common  watch  vibrates  5  times  in  a  second ;  if 
from  any  cause  each  vibration  is  gg1^  IGSS  tnan  i*s  proper 
time,  how  much  will  the  watch  gain  per  day? 

»  See  Section  24. 


§26.]  MISCELLANEOUS  EXAMPLES.  133 

25.  A  cloak  containing  6fyd.  of  broadcloth  that  is  l£yd. 
wide,  is  to  be  lined  with  silk  that  is  lyd.  wide.     How  much 
silk  will  be  required  ? 

26.  A  parlor  35ft.  6in.  long,  and  19ft.  6in.  wide,  is  to  be 
carpeted.     How  much  carpeting,  that  is  a  yard  wide,  will 
be  required,  provided  there  is  no  waste  in  matching  the 
figures  ?     How  much  will  be  required,  if  iyd.  is  lost  in 
matching  each  breadth  ? 

27.  A  man  wishes  to  build  in  his  cellar  a  potato  bin  that 
will  hold  40  bushels.     It  is  to  be  6ft.  long,  and  3ft.  6  in. 
high.     What  must  be.  the  width  of  the  bin  ? 

28.  If  a  family  of  3  persons  consume  a  barrel  of  flour  in 
11  weeks,  what  is  the  average  amount  of  bread  eaten  daily 
by  each  person,  3  pounds  of  flour  being  sufficient  to  make 
4  pounds  of  bread  ?  Ans.  1-J|  Ib. 

29.  Required  the  entire  quantity,  and  the  value  at  22cts. 
a  gallon,  of  the  milk  taken  by  D.  N.  Breed,  of  Lynn,  Mass., 
from  one  cow,  in  11  months  of  1839-40.     The  daily  average 
was  as  follows :— from  April  15  to  April  30,  1839,  6qt. ; 
in  May,  14qt. ;  in  June,  16qt. ;  in  July,  13qt. ;  in  August, 
12qt.;    in  Sept.,  llqt. ;    in  Oct.,  lOqt. ;  in  Nov.,  lOqt.; 
in  Dec.,  9qt. ;    in  Jan.,  1840,  9qt.  j    in  Feb.,  7qt.,  and 
from  March  1  to  March  15,  2qt. 

30.  Alfred  the  Great  had  large  candles  made  with  marks 
upon  them,  so  that  he  might  judge  of  the  time  by  the 
quantity  that  had  burned.1     If  a  candle  were  lighted  at 
noon,  of  such  size  that  only  9in.  would  be  consumed  in  24 
hours,  what  would  be  the  time  when  2£in.  had  burned  ? 

31.  How  much  coal,  at  2240  Ib.  to  the  ton,  would  fill  a 
bin  that  holds  160  bushels  of  potatoes  ? 

32.  How  many  wine  gallons  in  a  pail  12in.  in  diameter, 
and  14£in.  deep? 

*  Carpenter.    The  Clepsydra  had  not  been  introduced  into  England. 


134  THE  HOUSEHOLD.  [ART.  VII. 

33.  A  cylindrical  vessel  is  4  inches  in  diameter.     At 
what  heights  must  marks  be  placed  for  Iqt.  wine  measure ; 
for  Ipt.  beer  measure ;  and  for  1  half-peck  dry  measure  ? 

34.  Required  the  contents  in  wine  measure,  of  a  water- 
pail,  the  height  being  8|in.,  the  top  diameter  Him.,  and 
the  bottom  diameter  9in. 

35.  How  many  rolls  of  paper  hangings,  each  9yd.  long, 
and  £yd.  wide,  would  be  required  to  paper  one  side  of  a  room 
that  is  19ft.  long  and  9ft.  Gin.  high? 

36.  A  saving  of  $1  per  annum,  invested  at  6  per  cent, 
compound  interest,  will  amount  in  40  years  to  $154.761966. 
If  a  young  man  at  20  years  of  age,  commences  laying  up 
6  Jets,  per  day,  how  much  will  he  be  worth  when  he  is  60 
years  old  ? 

37.  A  man  expends  all  his  income,  but  in  looking  over 
his  accounts  at  the  end  of  the  year,  he  finds  that  $125  has 
been  laid  out  foolishly.     If  he  resolves  to  retrench,  and 
saves  that  amount  annually,  what  will  he  be  worth  in  40 
years  ? 

38.  Find  the  ratio  of  illumination  between  two  lights, 
which  throw  shadows  of  equal  intensity,  if  placed  at  the 
distances  of  9ft.  and  5£ft.  respectively.8 

*  "  The  following  method  of  measuring  the  comparative  illuminating 
power  of  different  lights,  is  founded  on  the  law  that  the  amount  of  rays 
thrown  on  a  given  surface,  is  inversely  as  the  square  of  the  distance 
of  the  illuminating  body.  Place  two  lights,  which  are  to  be  compared 
with  each  other,  at  the  distance  of  a  few  feet,  or  yards,  from  a  screen 
of  white  paper,  or  a  white  wall.  On  holding  a  small  card  near  the 
wall,  two  shadows  will  be  projected  on  it.  Bring  the  fainter  light 
nearer  the  card,  or  remove  the  brighter  light  farther  from  it,  till  both 
shadows  acquire  the  same  intensity.  Measure  now  the  distances  of 
the  two  lights  from  the  wall  or  screen,  and  the  squares  of  these  dis 
tances  will  give  the  ratio  of  illumination.  In  this  experiment  the 
spectator  should  be  equidistant  from  each  shadow." — Bigelow's 
Technology. 


§26.]  MISCELLANEOUS   EXAMPLES.  135 

39.  If  a  cylindrical  cistern  is  7ft.  Gin.  in  diameter,  what 
must  be  its  depth,  to  hold  45  hogsheads?     To  hold  30 
hogsheads  ? 

40.  If  wood  is  sawed  2ft.  long,  how  many  cords  will  thire 
be  in  a  pile  50ft.  long,  and  8ft.  3in.  high  ? 

41.  How  many  potatoes  are  there  in  a  cellar,  there  being 
5  barrels  full,  and  one  half-full,  each  barrel  holding  2bu. 
3pk.,  and  a  bin  that  is  10ft.  long  and  5ft.  wide,  being  filled 
to  the  height  of  3ft.  Gin. 

42.  An  imperial  gallon  of  sperm  oil,  burned  in  an  Argand 
lamp,    which  yields  a  light  equivalent  to  5  candles,  (6  to  a 
Ib.)  will    burn  about  100   hours.     The  solar  lamp,  with 
an  imperial  gallon  of  whale  oil,  yielding  a  light  equivalent 
to  4f  candles,  (6  to  a  Ib.)  will  burn  about  90  hours.*    When 
sperm  oil  is  $1.25  per  gallon,  and  whale  oil  62 Jets,  per 
gallon,  how  much  can  be  saved  on  every  gallon  by  the  use 
of  whale  oil  ?  Ans.  44|  cents. 

43.  A  cylindrical  cup  contains  3qt.  3.4gi.  wine  measure. 
Required  the  height,  the  diameter  being  6J  inches. 

Ans.  6  inches. 

44.  What  is  the  greater  diameter  of  a  pan,  the  less  dia 
meter  being  9iu.,  the  height  5in.,  and  the  contents  19.42pt. 
beer  measure  ?  Ans.   17  inches. 

45.  A  cylindrical  cistern,  6ft.  9in.  deep,  contains  35.43 
hogsheads.     What  is  its  diameter  ? 

46.  A  cylindrical  cistern,  7Jft.  in  diameter,  holds  47.24 
hogsheads.     What  is  its  depth  ?  Ans.  9ft. 

47.  What  should  be  the  diameter  of  a  cylinder  that  is 
6 Jin.  deep,  to  hold  a  half-bushel?  Ans.  14J  inches. 

48.  What  should  be  the  diameter  of  a  cylinder  that  is 
2 Jin.  deep,  to  hold  a  half-pint,  beer  measure? 

Ans.  Sin.  nearly. 

*  Parnell. 


136  ARTIFICERS'  WORK.  [ART.  VET. 

VIII.    ARTIFICERS'  WORK/ 
27.    THE  CARPENTER  AND  JOINER. 

1.  The  contents  of  a  board  are  found  by  multiplying  the 
length  by  the  mean  breadth,  provided  the  thickness  does 
not  exceed  1  inch.     If  the  board  tapers  regularly,  the  mean 
breadth  is  half  the  sum  of  the  two  end  breadths.     If  it  is 
of  irregular  shape,  the  breadth  should  be  taken  at  a  number 
of  different  points  at  equal  intervals,  and   their   average 
should  be  regarded  as  the  mean  breadth.     If  the  thickness 
exceeds  1  inch,  multiply  the  number  of  feet  in  the  area  by 
the  number  of  inches  in  the  thickness. 

If  the  contents  are  given,  and  either  of  the  dimensions 
are  required,  divide  the  contents  by  the  product  of  the  given 
dimensions. 

Examples. — A  board  12ft.  Gin.  long,  1ft.  Sin.  broad,  and 
fin.  thick,  contains  12 £x  lj  =  15|ft.  board  measure. 

A  board  13ft.  4in.  long,  1ft.  Sin.  broad,  and  1  \  in.  thick, 
contains  13JX  If  X  1^  =  33 Jft.  board  measure. 

2.  The  contents  of  square  or  hewn  timber  are  found  by 
multiplying  the  mean  breadth  by  the  mean  thickness,  and 
their   product   by   the    length.     The   mean    breadth    and 
thickness  are  found  in  the  same  manner  as  in  measuring 
boards.     Sometimes  the  contents  are  found  by  squaring  £ 
of  the  girt,  and  multiplying  by  the  length.     This  method 
is  erroneous,  and  always  gives  the  contents  too  great.     Tim 
ber  is  often  sold  by  board  measure.    Cubic  feet  can  be  reduced 
to  board  feet  by  multiplying  by  12,  (provided  the  thickness 
exceeds  1  inch,)  or  the  contents  can  be  found  at  once  in 
board  measure,  by  taking  one  of  the  dimensions  in  inches, 
and  the  other  two  in  feet.    If  the  contents  are  given,  and  any 
two  of  the  dimensions,  the  other  dimension  may  be  found, 

*  Ingram,  Gillespie,  Crossley  arid  Martin,  Nicholson,  Pratt,  and 
private  information. 


§27.]  THE   CARPENTER  AND   JOINER.  137 

by  dividing  the  contents  by  the  product  of  both  the  given 
dimensions. 

Examples. — A  hewn  log  19ft.  Gin.  long,  2ft.  2in.  broad, 
and  2ft.  lin.  thick,  contains  19 £  X  2$  X  2T15=88415  c.  ft.,  or 
10561ft.  board  measure. 

A  hewn  log  30ft.  Sin.  long,  1ft.  lOin.  wide,  and  1ft.  3in. 
thick,  contains  30f  X  22  X  H  =  168f  board  feet. 

3.  The  contents  of  round  timber  are  usually  found  by 
squaring   \  of  the  mean  girt,  and  multiplying  it  by  the 
length.     If  the  tree  is  covered  with  bark,  lin.   should  be 
deducted  from  the  quarter-girt  before  squaring.     If  the  bark 
is  very  thick,  more  than  lin.  is  sometimes  allowed.     No 
rough  timber  is  considered  measurable,  if  the  diameter  is 
less  than  6  inches.* 

This  rule  gives  the  contents  too  small,  nearly  in  the  pro 
portion  of  11  to  14.  A  ton  of  timber  is  considered  as 
equivalent  to  40  c.  ft.,  but  40  c.  ft.  of  round  timber  as 
generally  measured,  really  contains  50  c.  ft.  The  statement 
usually  given  without  any  explanation  in  Arithmetical  tables, 
that  "  40ft.  of  round  timber,  or  50ft.  of  hewn  timber  make 
1  ton,"  is  therefore  erroneous.  The  allowance  was  originally 
introduced  as  a  partial  compensation  to  the  purchaser  of 
round  timber,  for  the  waste  occasioned  in  squaring  it. 

If  the  true  content  is  required,  it  can  be  found  very 
nearly,  by  squaring  J  of  the  girt,  and  multiplying  by  twice 
the  length. b 

Example. — A  log  47ft.  Sin.  long,  and  girting  at  the 
ends  18ft.  and  6ft.,  has  for  its  mean  quarter  girt  1  of  (18  + 
6) -r-2  =  3.  Its  contents  are  therefore  32x  471 =429  c.  ft. 

4.  Flooring,  partitioning,   and  roofing,  and    all    large 
and  plain  work,  in  which  a  uniform  quantity  of  materials 

a  Let  G  be  the  mean  girt  in  feet,  L  the  length,  and  C  the  contents 
inc.  ft.  Then  L  =  166 -j-G2  ;  G  =  4xV(Ti"L. 

b  In  employing  this  rule,  as  well  as  in  the  usual  method,  lin.  should 
be  deducted  from  the  girt  if  the  tree  is  covered  with  bark. 

31TY    I 

V  OF  J 


138  ARTIFICERS'  WORK.  [ART.  VITI. 

and  labor  is  expended,  are  generally  measured  by  the 
"  square"  =  100  sq.  ft.  Some  work  is  measured  by  the 
linear  foot  or  yard,  some  by  the  square  foot  or  yard,  and 
some  by  the  cubic  foot.  For  some  of  the  more  difficult 
kinds  of  work,  it  is  usual  to  allow  "  measure  and  half,"  or 
"  double  measure,"  but  the  custom  varies  so  much  in  different 
places,  that  it  is  impossible  to  give  any  general  rules  of 
measurement.  Shingles  are  generally  18  inches  long,  and 
of  the  average  width  of  4  inches.  When  nailed  to  the  roof 
i  is  usually  left  out  to  the  weather,  and  6  shingles  are  there 
fore  required  to  a  square  foot.  But  on  account  of  waste 
and  defects,  1000  should  be  allowed  to  a  "  square."  The 
weight  of  a  square  of  partitioning  may  be  estimated  at  from 
1500  to  2000  Ib. ;  a  square  of  single-joisted  flooring,  at  from 
1200  to  2000  Ib. ;  a  square  of  framed  flooring,  at  from  2700 
to  4500  Ib.  \  a  square  of  deafening,  at  about  1500  lb.a  When 
a  floor  is  covered  with  people,  120  Ib.  per  sq.  ft.  should  be 
added  to  the  weight. b 

5.  The  sliding  rule  is  often  used  by  carpenters  and  other 
artificers,  in  the  measurement  of  timber  and  work.  The 
foot  is  divided  in  the  usual  way,  into  inches  and  eighths  of 
an  inch,  and  it  is  also  subdivided  decimally  and  logarithmi 
cally,  so  as  to  facilitate  the  labor  of  computing.  The  use 
of  the  rule  can  only  be  learned  by  practice. 

The  carpenter's  square  is  used  in  determining  whether  the 
corners  of  boards  or  buildings  are  square.  In  framing 
buildings,  the  corners  are  sometimes  "  squared,"  by  measur 
ing  8ft.  on  one  timber,  and  6ft.  on  the  other,  and  placing 
the  extremities  of  the  measured  lines  10ft.  apart.  This 
mode  of  operation  is  founded  on  the  property  of  right-angled 
triangles,  that  the  square  of  the  hypothenuse  is  equal  to  the 
sum  of  the  squares  of  the  other  two  sides.  A  roof  is  said 
to  have  a  true  pitch,  when  the  length  of  each  rafter  is  f  of 

a  Hatfield.  b  Tredgold. 


§27.]  THE   CARPENTER   AND   JOINER.  139 

the  breadth  of  the  building.     The  two  sides  of  the  roof  then 
form  nearly  a  right  angle. 

6.  The  area  of  posts. — All  rules  for  determining  the 
resistance  of  timber,  should  be  based  on  the  supposition 
that  the  timber  is  of  "merchantable"  quality,  straight- 
grained,  seasoned,  and  free  from  large  knots,  splits,  decay, 
or  other  defects.  When  the  height  of  a  piece  of  timber 
exceeds  about  ten  times  its  thickness,  it  will  bend  before 
crushing.  To  find  the  area  of  a  post  that  will  safely  bear 
a  given  weight,  when  the  height  of  the  post  is  less  than 
ten  times  its  least  thickness ; — Divide  the  given  weight  in 
pounds  by  1000  for  pine,  or  by  1400  for  oak,  and  the  quotient 
will  be  the  least  area  of  the  post  in  inches. 

EXAMPLES. 

1.  Required  the  contents  of  a  board  13ft.  Tin.  long,  fin. 
thick,  and  1ft.  Gin.  wide,  and  its  value  at  9  Jets,  per  foot. 

Partial  Ans.  Value  $1.94. 

2.  The  length  of  aboard  is  lift.  8in.,  the  thickness  IJin., 
and  the  breadths  measured  at  five  different  points  are  as  fol 
lows;  1ft.  6in.,  2ft.  3in.,  1ft.  9in.,  2ft.  6in.,  and  2ft.     What 
are  the  contents  ?  Ans.  35ft. 

3.  How  many  cubic  feet,  and  how  many  feet  board  mea 
sure,  in  a  log  21ft.  Gin.  long,  the  mean  breadth  being  1ft. 
4in.,  and  the  mean  thickness  1ft.  3in.  ? 

4.  A  log  is  42ft.  9in.  long,  and  the  girts  at  four  points, 
outside  of  the  bark,  are  56,  45 £,  58,  and  64 }  inches.     What 
are  its  contents  by  the  ordinary  rule,  and  by  the  correct 
rule? 

5.  How  many  squares  of  flooring  in  a  four  story  house, 
42ft.  Gin.  by  28ft  4in.  within  the  walls,  deducting  from  each 
floor  the  vacancy  for  the  stairway,  13ft.   by  7ft.  Gin. ;  and 
what  is  the  cost  of  the  whole,  at  $3.87£  per  square? 

6.  A  board  measures  at  five  different  points,  1ft.  Gin., 


140  ARTIFICERS'  WORK.  [ART.  vm. 

1ft.  9in.,  1ft.  8in.,  1ft.  4in.,  and  1ft.  3in.,  in  breadth.    What 
is  its  length,  the  area  being  18fft.  ?  Ans.  12ft.  6'. 

7.  A  stick  of  timber  9ft.  6in.  long,  measures  1  ton  14| 
c.  ft.     What  is  its  mean  area  ?  Ans.  5ft.  9'. 


38.     THE  MASON. 

1.  Rubble*  walls  are  generally  measured  by  the  perch, 
which  is  16  Kt.  long,  1ft.  deep,  and  IJft.  thick,  and   is 
therefore  equivalent  to  24 f  c.  ft.     In  some  places,  25  c.  ft. 
is  allowed  to  the  perch,  in  measuring  stone  before  it  is  laid, 
and  22  c.  ft.  after  it  is  laid  in  the  wall.     When  the  wall  is 
not  of  uniform  height,  the  height  should  be  measured  at 
several  places,  from  the  bottom  of  the  foundation  to  the  top 
of  the  wall,  and  the  mean  height  employed  in  computing 
the  solid  contents.     Nine  pecks  of  good  lime  and  3  one 
horse  loads  of  sand,  will  make  mortar  for  3  perches  of  wall. 
Rough  stone  and  marble  are  often  measured  by  the  cubic 
foot.     In  measuring  workmanship,  linear  feet  and  yards,  and 
square  feet  and  yards  are  employed. 

2.  The  rood  of  36  sq.  yd.   is  sometimes  employed   in 
measuring  walls  that  are  more  than  18in.  thick.     The  wall 
should  first  be  reduced  to  2ft.  thick.     Thus  a  wall  90ft.  long, 
8ft.  high,  and  21in.  thick,  is  equivalent  to  i  of  a  wall  90ft. 
long,  8ft.  high,  and  2ft.  thick;  1  x  90  X  8=630  sq.  ft;  630-7- 
9=70  sq.  yd.;  70-r36=l  rood  34yd. 

3.  Cisterns  can  be  measured  accurately  by  finding  the  solid 
contents  in  cubic  inches,  and  dividing  by  the  number  of 
cubic  inches  in  a  hogshead,  (63  X  231.)     But,  in  measuring 
circular  cisterns,  the  rules  given  in  Sect.  25,  are  much  more 
convenient  than  this  method,  and  are  sufficiently  correct  for 
ordinary  purposes.     If  great  accuracy  is  required,  measure 
the  diameter  and  depth  in  inches,  multiply  the  square  of  the 

*  Rough  stone  work  is  called  rubble  work. 


§28.]  THE   MASON.  141 

diameter  by  the  depth,  and  multiply  the  product  by  either 
of  the  following  numbers,  to  obtain  the  contents  : — 


In  cubic  feet 
.00045451 

In  ale  gallons 
.002785 

In  wine  gallons 
.0034 

In  cubic  inches 
.785398 

In  hogsheads 
.000054 

In  bushels 
3TSJT 

In  Ibs.  of  water 
.028326 

In  Imperial  gal. 
.0028326 

4.  Arches  are  measured  by  applying  a  line  close  to  the 
surface  in  taking  the  dimensions.  If  the  arch  is  not  of 
uniform  length,  breadth,  and  thickness,  the  dimeLsions  may 
be  measured  at  several  points,  and  the  mean  of  all  the 
measurements  taken.  A  pointed  arch  will  sustain  almost 
any  weight  on  its  crown,  provided  the  lowest  stones  do  not 
give  way.  Therefore  the  Gothic  arch  is  stronger  for  lofty 
buildings  than  the  circular,  but  the  circular  arch  is  far  better 
adapted  than  the  Gothic,  for  bridges  or  other  works,  where 
every  part  of  the  arch  may  be  exposed  to  equal,  or  nearly 
equal  pressures. 

EXAMPLES. 

1.  How  many  cubic  feet  in  a  block  of  marble,  4ft.  6in. 
long,  3ft.  Sin.  wide,  and  2ft.  4in.  thick  ? 

2.  How  many  perches  of  24f  c.  ft.  in  a  wall  97ft.  long, 
and  2ft.  thick,  the  heights  at  five  different  points  being  4ft. 
8in.,  3ft.  8in.,  3ft.  9in.,  4ft.,  and  5ft.  2in.  and  how  much 
lime  will  be  required  to  make  mortar  for  the  whole  ? 

3.  Required  the  cost  at  50cts.  per  perch  of  25  c.  ft.,  of 
making  a  cellar  wall  6ft.  3in.  high,  and  1ft.  Gin.  thick,  the 
outside  measurement  being  47ft.  long,  and  22ft.  Gin.  wide  ? 

4.  How  many  hogsheads  in  a  cylindrical  cistern  13ft.  Gin. 
in  diameter,  and  lift.  7in.  deep  ? 

5.  How  much  hewn  work  in  18  lintels  and  sills,  each4ffc. 
by  1ft.  Gin.,  and  in  chimney  coping,  58ft.  Gin.  by  19in.,  and 
what  is  the  cost  of  the  whole  at  10£cts.  per  foot  ? 

Partial  ATIS.  Cost,  $21.06. 


142  ARTIFICERS'  WORK.  [ART.  vm. 

6.  Find  the  contents  in  cubic  feet,  in  hogsheads,  and  in 
Imperial  gallons,  of  a  cylindrical  cistern  10ft.  6in.  in  dia 
meter,  and  9ft.  2in.  deep. 

1st  Ans.  793.738  c.  ft. 


.    THE  BRICKLAYER. 

1.  Brickwork  is  measured  either  by  the  square  yard,  or 
by  the  square  rod,  and  is  usually  estimated  at  1J  bricks 
thick,  =  12  inches,  the  actual  thickness  being  reduced  to  the 
standard  of  1^  bricks.     272  sq.  ft.  is  generally  counted  as 
a  rod,  the  fraction  of  i  sq.  ft.  being  rejected.*     The  rood 
of  18ft.  sq.,  or  324  sq.  ft.,  and  the  rod  of  16£  sq.  ft.,  are 
also  used.     But  the  common  practice  is  now  to  reckon  bricks 
by  the  1000,  the  number  required  depending  on  the  size  of 
the  bricks.     In  estimating  the  number,  an  allowance  of  Jj 
of  the  solid  contents  should  be  made  for  the  space  occupied 
by  the  mortar. 

2.  The  usual  dimensions  of  bricks,  are;    length  8in., 
breadth  4in.,  thickness  2in.     Whatever  the  length  may  be, 
the  breadth  is  generally  ?,  and  the  thickness  i  as  great; 
thus  a  brick  9  inches  long,  would  be4z  in.  wide,  and  2i  in. 
thick.     In  some  places,  all  walls  are  charged  as  solid,  no 
allowance  being  made  for  doors,  windows,  or  other  openings  ; 
in  others,  the  openings  are  deducted  in  charging  for  mate 
rials,  but  the  workmanship  is  estimated  as  if  the  walls  were 
solid  ;  in  others  an  allowance  of  one  half  is  made  for  all 
openings  ;  and  in  others  the  actual  materials  employed,  and 
workmanship  expended,  are  charged. 

3.  The  number  of  rods  in  any  wall  may  be  found  by 
multiplying  the  area  of  the  surface  by  i  of  the  number  of 
half-bricksb  in  the  thickness,  and  dividing  the  product  by  the 
number  of  square  feet  allowed  to  a  rod.     A  load  of  sand= 
30  struck  bushels  ;  a  ton  =  24  c.  ft. 

*  The  weight  of  a  rod  of  brickwork  may  be  estimated  at  16  tons. 
b  A  half-brick  =  4  inches. 


§30.]  THE  PLASTERER.  143 

EXAMPLES. 

1.  How  many  bricks  of  the  usual  size,  will  be  required 
to  make  a  wall  40ft.  long,  16ft.  high,  and  2  bricks  thick, 
making  allowance  for  mortar  ? 

Ans.  ^  of  40  X16XHX  27=20736. 

2.  How  many  roods  of  324ft.  in  a  wall  96ft.  long,  10ft. 
high,  and  3  bricks  thick,  and  what  is  the  cost  of  the  bricks 
at  $7.75  per  M?  Ans.  Cost,  $361.58. 

3.  A  garden  contains  1J  acres,  and  is  150ft.  wide.     Re 
quired  the  cost  of  enclosing  it  with  a  brick  wall  9ft.  4in. 
high  and  3  bricks  thick,  at  $7.50  per  M.,  deducting  2  doors, 
each  6ft.  3in.  by  4ft.,  and  a  gateway  12ft.  wide  ? 

Ans.  $3404.19. 

3O.    THE  PLASTERER. 

Plain  plastering  is  measured  either  by  the  square  foot, 
the  square  yard,  or  the  "square"  of  100  sq.  ft.  The  number 
of  coats,  and  the  quality  of  the  finishings,  should  be  stated 
in  the  bill.  Cornices  and  mouldings,  if  12  inches  or  more 
in  girt,  are  sometimes  estimated  by  the  square  foot;  if  less 
than  12  inches,  they  are  usually  measured  by  the  linear  foot. 
Plastering  on  walls  is  called  rendering  ;  ceiling,  is  plastering 
on  laths.  The  custom  varies  as  to  the  proper  allowance 
for  doors  and  windows. 

EXAMPLES. 

1.  What  will  be  the  cost  of  plastering  a  ceiling  21ft.  6in. 
long,  and  19ft.  6in.  wide,  at  llcts.  per  sq.  yd.  ? 

2.  How  much  plastering  on  a  partition  22ft.  3in.  long, 
and  7ft.  9in.  high,  deducting  two  doors,  each  6ft.  by  3ft. 
2in.,  and  what  will  it  cost  at  12£cts.  per  sq.  yd.  ? 

Ans.  Cost,  $1.87. 

3.  How  many  squares  of  ceiling,  and  of  rendering,  and 


144  ARTIFICERS'  WORK.  [ART.  vin. 

how  many  feet  of  cornice,  in  a  hall  60ft.  long,  28ft.  Gin. 
wide,  and  10ft.  high,  deducting  17yd.  5ft.  6'  for  doors  and 
windows?  Ans.  16  sq.  lyd.  2ft.  67  of  rendering. 

17  sq.  lyd.  1ft.  of  ceiling. 

177ft.  of  cornice. 

31.    THE  PAINTER  AND  GLAZIER. 

Painting  is  measured  by  the  square  yard.  In  taking  the 
dimensions,  the  measuring  line  is  laid  into  all  the  mouldings, 
BO  as  to  reach  every  point  which  the  brush  touches. 

Glazing  is  sometimes  measured  by  the  square  foot,  some 
times  by  the  piece,  or  by  the  light.  In  estimating  by  the 
square  foot,  it  is  customary  to  include  the  whole  sash.  Cir 
cular  or  oval  windows  are  measured  as  if  they  were  square. 

EXAMPLES. 

1.  A  room  is  24ft.  long,  18ft.  6in.  wide,  and  9ft.  6in. 
high.     How  many  yards  of  painting  are  in  it,  deducting  a 
fire-place  4ft.  6in.  by  4ft.,  and  3  windows,  each  6ft.  by  3ft. 
3in.  ?  Ans.  81yd.  2ft. 

2.  At  12  Jets,  per  yard,  what  will  it  cost  to  paint  a  wall 
15ft.  8'  by  8ft.  3'?  Ans.  $1.78. 

3.  How  many  feet  of  glazing  in  an  oval  window  4ft.  6' 
by  2ft.  8'  ? 

4.  At  18fcts.  per  foot,  what  will  it  cost  to  glaze  3  stories 
of  a  house,  with  8  windows  in  each  story,  the  breadth  of 
each  window  being  3ft.,  and  the  height  5ft.  4in.  ? 

Ans.  $72. 

33.     THE  PAVER,  SLATER  AND  TILER. 

Paving  is  measured  by  the  square  foot,  the  square  yard, 
or  the  rood  of  36  sq.  yds.  If  the  pavement  is  grooved,  the 
grooves  are  added  to  the  surface  measure. 

Slating  and  Tiling  are  measured  by  the  square  yard,  by 


§33.]  THE   PLUMBER.  145 

the  rood,  or  by  the  "square"  of  100  sq.  ft.  It  is  not  usual 
to  make  any  deductions  for  chimneys,  skylights,  or  other 
apertures.  In  measuring  the  girt  of  slate  roofs,  allowance 
must  be  made  for  the  double  row  at  the  eaves.* 

EXAMPLES. 

1.  A  yard  is  80ft.  long,  and  28ft.  Gin.  wide.     What  will 
it  cost  to  pave  it  at  TOcts.  per  square  yard  ? 

2.  The  side  of  a  square  court  measures  120  feet.     "What 
will  it  cost  to  pave  it,  leaving  a  nagged  walk  6ft.  wide  around 
the  outside,  at  75cts.  per  yd.,  and  paving  the  rest  with 
bricks  at  $8.00  per  M.,  allowing  4  bricks  to  a  square  foot? 

Am.  8601.25. 

3.  How  many  roods  of  tiling  in  a  roof  52ft.  8'  long,  and 
45ft.  6' in  girt? 

4.  At  $1.30  per  yard,  what  will  be  the  expense  of  slating 
a  roof  49ft.  6'  long,  and  girting  46f  ft.  ?      Am.  83331. 

33.    THE  PLUMBER. 

Plumbers'  work  is  generally  done  by  the  pound  or  hun 
dred-weight.  A  square  foot  of  sheet  lead,  y^in.  thick,  weighs 
5.899  Ib.  From  this  value  the  weight  of  a  square  foot  of  any 
other  thickness  can  readily  be  determined.  Lead  pipes  of 
fin.  bore,  weigh  about  10  Ib.  per  yd.;  lin.  bore,  12  Ib. ; 
Hm.  bore,  16  Ib.;  IJin.  bore,  18  Ib.;  If  in.  bore,  21  Ib. ; 
Sin.  bore,  24  Ib. 

The  contents  of  lead  pipe  of  any  given  dimensions,  may 
be  found  by  the  table  in  Sect.  28,  for  measuring  cisterns. 

EXAMPLES. 

1.  What  is  the  weight  of  1  sq.  ft.  of  lead,  the  thickness 
being  £  of  an  inch  ? 

*  The  weight  of  a  square  foot  of  slating  may  be  estimated  at  Hi  Ih. 
The  greatest  force  of  the  wind  on  a  roof,  is  about  40  Ib.  per  square 
foot.--  Tredgold. 

10 


146  ARTIFICERS'  WORK.  [ART.  vm. 

2.  Find  the  weight  of  lead  necessary  to  cover  one  side  of 
a  roof  36ft.  9in.  long,  and  18ft.  Sin.  wide,  at  8£lb.  per 
square  foot. 

3.  What  is  the  cost  of  covering  and  guttering  a  roof,  at 
18s.  per  cwt.,  the  length  of  the  roof  being  43ft.,  and  the 
girt  32ft.;  57ft.  of  guttering  2ft.  wide;  weight  of  roofing, 
9.831  Ib.  per  sq.  ft.,  and  of  guttering  7.373  Ib.  per  sq.  ft.  ? 

Ans.  £115  9s.  lid. 

4.  Required  the  expense  of  a  leaden  pipe,  If  in.  bore, 
and  185ft.  long,  at  llcts.  per  Ib. 

5.  A  roof  40ft.  long,  and  57ft.  girt,  is  covered  with  lead 
^in.  thick;  the  water  pipe  is  llin.  bore  and  52ft.  long,  and 
the  waste  pipe  is  2in.  bore  and  40ft.  long;  the  water  cistern 
is  4ft.  3in.  long,  3ft.  6in.  wide,  and  3ft.  deep,  and  lined  with 
lead   Jin.  thick.     What  is  the  amount  of  the  plumber's 
bill,  rating  the  sheet  lead  at  $7.50  per  cwt.,  and  the  pipe  at 
lOcts.  per  pound  ? 

34.    SPECIFICATION  AND  ESTIMATES. 

1.  Specification  of  materials  to  be  provided,  and  labor  to  be  per 
formed,  in  the  construction  and  finishing  of  a  Schoolhouse  for  the 
City  of  Worcester,  to  be  erected  on  Summit  Street  in  said  City, 
according  to  plans  drawn  by  E.  Boyden,  Architect,  and  herewith 
presented. 

Size  of  House. — 58  feet  long  by  50  feet  wide,  not  including  the 
projection  of  the  pilasters. 

Height  of  stories  as  figured  on  Section  of  Front  Elevation. 

The  location  of  the  cellar,  and  depth  of  excavation,  to  be  deter 
mined  by  the  building  committee. 

A  well  to  be  dug  upon  the  lot  in  such  location  as  directed  by 
the  building  committee,  and  also  to  be  stoned  up  so  as  to  leave  a 
diameter  of  three  and  a  half  feet  in  the  clear. 

All  earth  dug  out  of  the  cellar  and  well,  to  be  deposited  upon 
the  lot,  as  may  be  directed  by  the  building  committee. 

The  foundation  walls  to  be  three  feet  thick  at  the  bottom,  2J 
feet  thick  at  the  top,  and  of  such  height  as  may  be  determined  by 


§34.]  SPECIFICATION  AND  ESTIMATES.  147 

the  building  committee.  The  walls  to  be  made  of  large  square 
block-stone,  well  faced  and  bonded.* 

Underpinning  2  feet  wide,  and  not  less  than  8  inches  thick,  to 
be  made  of  rough  split  South  Ledge  stone,  with  a  rough-hammered 
bevelled  washb  between  the  pilasters.  The  face  of  the  under 
pinning  to  project  as  far  forward  as  the  face  of  the  pilasters. 

Stone  steps,  of  fine-hammered  South  Ledge  granite,  located  and 
of  such  size  as  represented  on  the  plan. 

All  the  foundations  for  piers  and  partition  walls,  to  be  not  less 
than  8  inches  thick,  and  to  be  placed  as  represented  on  the  plan 
of  the  cellar. 

Stone  lintels  over  all  the  cellar  windows,  and  four  stone  thresh 
olds  of  fine-hammered  South  Ledge  granite,  to  be  made  of  the 
dimensions  marked  on  the  ground  plan. 

Outside  wall  of  brick  to  be  one  foot  thick,  with  pilasters  pro 
jecting  4  inches  beyond  the  face  of  the  wall,  and  Corbel-Course0 
and  Frieze,4  as  shown  on  elevation. 

Four  brick  piers  in  the  cellar,  each  one  foot  square,  and  a  brick 
partition  8  inches  thick  to  be  carried  from  the  bottom  of  the  cellar 
to  the  attic  floor,  as  represented  on  the  plans. 

The  chimneys  to  be  located  and  constructed  as  represented  on 
the  ground  plan. 

Building  to  be  lathed  throughout,  and  plastered  with  two  coats, 
except  the  play-room  in  the  basement.  Walls  not  to  be  plastered 
underneath  the  ceiling. 

All  the  windows  to  have  four-course,6  tooled  sandstone  caps, 
and  two-course  sills.  The  doors  to  have  five-course  caps  of  the 
same  material. 

Twenty  ventilating  registers,  each  one  foot  in  diameter,  to  be 
furnished  and  inserted,  two  in  each  ventilating  flue,  one  near  the 
floor  and  the  other  near  the  ceiling.  All  the  rooms  to  be  thus 
ventilated  except  the  play-room. 

For  the  arrangement  and  sizes  of  timber  in  the  first  three  floors, 
see  plan  of  flooring,  as  represented  on  the  basement  plan. 

a  Laid  like  bricks,  so  that  the  joints  will  not  come  over  each  other. 
b  The  wash  of  the  stone  is  the  inclined  surface  for  water  to  run  ofT. 
c  Projections  in  a  wall  to  sustain  the  timbers  of  a  floor  or  roof,  are 
called  corbels. 

d  The  part  of  the  wall  above  the  pilasters. 
c  Of  the  thickness  of  four  courses  of  brick. 


148  ARTIFICERS'  WORK.  [ART.  vm. 

Frames  to  be  made  entirely  of  good  spruce  framing  timber. 
Joists  in  4th  or  attic  floor  to  be  2  X  9,  framed  15  inches  between 
centres.  All  other  timber  in  roof  and  observatory  to  be  of  the 
size  figured  on  the  plan  of  roof. 

Joists  in  first  three  floors  to  be  jointed  fin.  crowning*  to  15 
feet  in  length.  All  floorings  to  be  bridged  with  good  X  bridging 
where  marked  on  plan  of  flooring.  All  large  timber  to  be  well  and 
properly  secured  to  the  brick  walls  by  suitable  anchor  irons.  The 
joists  to  be  also  secured  in  a  similar  manner,  as  often  as  once  in 
every  ten  feet. 

Roof  to  be  boarded  with  suitable  fin.  boards,  planed,  jointed, 
matched,  and  suitably  tinned.  Roof  to  be  bracketed  and  project 
as  represented  on  elevation.  Observatory  to  be  framed  and  finished 
in  every  respect  as  represented  and  figured  on  plan  of  observatory, 
front  elevation,  and  plan  of  roof. 

Lintels  G  x  7  Jin.  to  be  furnished  for  all  windows  and  doors. 
All  floors  to  be  lined  with  suitable  fin.  lining  boards  laid  edge  to 
edge,  and  nailed  with  8d.  nails. 

The  floor  to  the  upper  schoolroom  to  be  deafened  in  the  centre 
of  the  floor  joists  with  a  suitable  coat  of  coarse  mortar.  All  top 
floors  to  be  of  suitable  southern  hard  pine  |in.  thick,  and  not 
to  exceed  Gin.  in  width,  well  laid  and  nailed  with  12d.  floor  nails. 

Five  iron  columns  to  be  placed  as  represented  on  plans  of  base 
ment  and  second  story,  of  size  and  quality  like  those  in  the  Pleas 
ant  Street  Schoolhouse. 

Partitions  to  be  arranged  as  represented  on  plans.  Those  in 
basement  and  2d  story,  to  be  constructed  of  2x5in.  partition 
plank,  bridged  once  with  1|-  X  5in.  herring-bone  bridging.  Those 
in  3d  story  to  be  of  2  X  6in.  partition  plank,  bridged  twice  with 
herring-bone  bridging  as  above.  All  partition  plank  to  be  jointed 
and  set  edgewise,  so  as  not  to  exceed  one  foot  between  centres. 
The  contractor  is  also  to  make  all  necessary  arrangements  in  the 
partitions  for  pipes  to  convey  heated  air  to  all  the  different  apart 
ments,  and  to  do  all  necessary  wood-work  preparatory  to  putting 
in  registers  to  admit  the  hot  air.  The  house  to  be  furred  through 
out  with  f  x2£  inch  furrings,  placed  at  the  distance  of  one  foot 
between  their  centres. 

Teachers'  platforms  to  be  elevated  6  inches  above  the  floor,  of 
the  situation  and  sizes  represented  on  plans. 

a  Long  timbers  are  usually  made  "  crowning"  in  the  centre,  so  as  to 
allow  for  settling. 


§34.]  SPECIFICATION   AND   ESTIMATES.  149 

All  window-frames  to  be  constructed  as  represented  by  the 
drawings,  with  hard  pine  pulley  styles.1 

Four  cellar  window  frames,  to  be  made  of  2in.  chestnut  plank, 
each  large  enough  for  a  window  with  4  lights  of  9 X 12  glass.  All 
sash  to  be  of  first  quality  Eastern  pine  stock,  lip  sash,  ogeeb  style, 
l^in.  thick,  to  be  double-hung  with  suitable  weights,  cords,  and 
pulleys,  and  glazed  with  best  quality  German  glass  of  such  sizes 
as  figured  on  front  elevation. 

All  windows  in  2d  and  3d  stories  to  have  blinds  to  slide  into 
the  walls  upon  each  side  of  the  window,  as  shown  on  detail  of 
window  frame. 

Four  outside  doors  with  side  lights,  as  represented  on  front  ele 
vation,  each  8ft.  high  by  3Jft.  wide,  and  2in.  thick,  4  panels  with 
bevel  joints,  to  be  hung  with  3  sets  4in.  loose  joint  butts,  and 
trimmed  with  suitable  mineral  knobs.  The  two  front  doors  to 
have  suitable  mortice  locks,  and  the  two  other  doors  suitable  bolts 
inside.  All  other  doors  to  be  3^x"i^-  ^  Pane^s>  If  in.  thick, 
with  bevel  joints,  suitably  hung  with  3  sets  of  4in.  loose  joint 
butts,  and  trimmed  with  suitable  mineral  knobs  and  mortice  locks. 

A  flight  of  cellar  stairs,  with  hard  pine  treads  lin.  thick,  placed 
as  represented  on  cellar  plan.  All  other  stairs  located  as  repre 
sented  on  plans,  with  hard  pine  risers  lin.  thick,  and  treads  l£in. 
thick,  and  to  have  cherry  newels0  and  hand-rails.  All  staircases 
to  be  ceiled  up  on  the  well-room d  side  as  high  as  the  hand-rail, 
and  all  rooms  to  be  ceiled  as  high  as  the  window  stools,  with  suit 
able  Eastern  pine  stock  fin.  thick,  not  to  exceed  6  inches  in  width, 
jointed,  matched,  and  beaded. 

Cleats  to  be  put  up  in  entries  and  recitation  rooms,  sufficient  to 
contain  30  doz.  glazed  clothes-hooks,  placed  6  inches  apart,  and 
to  be  provided  with  said  hooks. 

All  doors  and  standing  wood-work  inside,  to  be  grained  in  imi 
tation  of  oak.  Jet  and  brackets  to  be  painted  and  sanded  in  imi 
tation  of  sandstone.  Observatory,  window  frames,  and  door  frames, 
to  be  painted  white  and  sanded.  All  painting  to  be  done  with 
three  coats  of  pure  white  lead  and  linseed  oil,  colored  as  above 
specified.  Tinning  upon  the  roof  to  be  painted  with  one  coat  upon 
the  under  side,  and  two  upon  the  upper  side,  of  spruce  yellow  and 
boiled  linseed  oil. 

a  The  strips  in  which  the  window  pulleys  are  placed. 
b  An  ogee  is  a  moulding  resembling  the  letter  S  in  its  outline. 
c  The  posts  into  which  the  hand-rail  is  inserted  are  called  newels. 
d  The  space  occupied  by  the  stairway  is  called  the  well-room. 


150 


ARTIFICERS'  WORK. 


[ART.  vm 


A  piece  of  raised  tin  to  be  placed  on  each  side  of  each  front 
door  to  turn  the  water  off  outside  of  the  door.  A  single  floor  to 
be  laid  in  the  attic,  of  common  white  pine  boards  fin.  thick,  planed, 
jointed,  matched,  and  nailed  with  8d.  nails.* 


2.  ESTIMATED 


45  squares  of  roof  . 
33       "  ceiling 

32  windows     . 
12       «  . 

5  iron  columns 


at  $12 

.       8 

.     16 

.     10 

12 


4  outside  doors  with  frames  and  trimmings  16 


6  nights  of  stairs  . 
30  squares  attic  flooring  . 

3  hard  pine  floors  . 
Iron  work  for  building 
Observatory     . 
40  brackets 

14  inside  doors  trimmed  . 
14  door  casings 
35  window  casings  . 
2000ft.  boards  for  furring 
Labor  in  furring 
Painting 

32000ft.  timber  in  frame  . 
Deafening  $150.     Nails  $10 
2200yds.  plastering 
200000  brick 
Caps  and  sills 
175  seats 
175     «. 

125  perch  of  stone  . 
500yd.  excavation    . 


50 
3 

45 

2.25 
8 

4.75 
1.12J 
16 


900.00 


250.00 


75.00 
250.00 


22 


.22 

8.50 

3.25 
.80 

1.00 
.15 


225.00 


$7991:62* 


a  The  foregoing  specification  furnishes  materials  for  a  great  number 
of  useful  questions,  which  the  teacher  may  frame  so  as  to  adapt  them 
to  the  wants  of  his  pupils. 

b  In  each  item,  the  expense  of  labor  is  included. 


§  34.]  ESTIMATES.  151 

3.  ESTIMATE  OP  THE  MATERIALS  AND  LABOR 

REQUIRED  IN  A  COTTAGE. 

From  Ranletfs  Architect. 

$ 

296  cubic  yds.  excavation  @      .09 

2538  cubic  ft.  stone  work  @      .10 

7  stone  sills  @      .50 

24  linear  ft.  steps  @      .14 

Cistern  work  $3.50;  4  hearths  @    3.00 

1  marble  mantel  $50;  2  veined  mantels  @  25.00 

2  brown  stone  chimney  caps  @  14.00 
1451  square  yds.  plastering  @      .26 
36500  brick  @    9.50  per  M. 
268  linear  ft.  cornice  @      .24 
16196ft.  timber  in  frame  @    2.00  per  hund. 
474  joist,  set  in  frame  and  partitions       @      .18 

4782  sq.  ft.  of  sheathing  and  siding  @      .07 

2283      «  «        and  iron  roof  @      .16£ 

60  linear  ft.  3in.  leader,  @  12  Jc.,  114ft.  @      .11 

3485  square  ft.  of  interior  floor  @      .04 

1002      "       «         veranda    "  @      .08 

1184     «     '"  «        roof  @      .09 

172  lin.  ft.  main  cornice  @      .85 

83     «     «  wing       «  @      .70 

199  «     «  veranda  "  @      .55 

15  veranda  columns @  $10.50 ;  3  antse  @    3.00 

164ft.  veranda  cornice  and  filling  @      .14 

15  steps  and  rises — back  stairs  @    1.00 

18     "       «      «  principal  "  @    3.50 

2  front  doors,  with  side  and  head  lights  @  30.00 

11  doors  in  principal  story  @  11.00 

10     "          second         "  @    8.00 

10     "          wing  @    7.00 

7  double  windows,  first  story  @  14.00 

8  "  «      second  "  @  10.00 


152                            STRENGTH   OF   MATERIALS.  [ART.  IX. 

$ 

7  double  windows,  wing                           @  9.00 

8  single        "                                           @  6.50 
6  cellar         "                                            @  2.50 

3  wood  mantels  @  $4.50;  7  bells           @  3.25 
400  square  yds.  tight  furring                    @  .07 
197  linear  ft.  of  blinds                             @  .80 
1152  "       "  of  base                               @  .04 
12  closets,  to  shelve  and  put  in  hooks     @  4.50 

1000  Ib.  white  lead  in  oil                          @  7.00  per  hund. 

43  gallons  linseed  oil                                @  80 

4  "           «       «  boiled                      @  .90 

5  spirits  turpentine                  @  .50 
i         "         varnish                                  @  4.00 
30  Ib.  putty,  @  4ct. ;  10  Ib.  litharge       @  .06 
3  Ib.  glue,  @  20ct. ;  2  Ib.  lampblack  in  oil  @  .40 

6  Ib.  chrome  yellow  in  oil                         @  .30 
60  days  painters'  labor                              @  1.75 
Hardware, — locks,  bolts,  window  weights,  &c.  99 .39 


$4550.00 


IX.    STEENGTH  OF  MATERIALS. 

ALL  solid  substances  may  be  exposed  to  four  kinds  of 
strains.  1st,  they  may  be  torn  asunder,  as  in  the  case  of 
ropes,  tie-beams,  king-posts,  &c.  2d,  they  may  be  crushed, 
as  in  the  case'of  columns,  posts,  &c.  3d,  they  may  be  broken 
across,  as  in  the  case  of  joists,  beams,  &c.  4th,  they  may  be 
twisted  or  wrenched,  as  in  the  case  of  wheel-axles,  the  screw 
of  a  press,  the  rudder  of  a  vessel,  &c.  Numerous  experi 
ments  have  been  made  to  determine  the  strength  to  which  the 
materials  in  common  use  may  safely  be  subjected,  and  tables 
have  been  compiled  from  the  results  of  those  experiments. 


§36.] 


STRENGTH   OP   TIMBER. 


153 


35.  TABLE  OF  THE  FLEXIBILITY  AND  STRENGTH 
or  TIMBER/ 


Name  of  wood. 

Specific 
Gravity. 

Value 
of  U.b 

Value  of  E.c 

Value 
ofS.d 

Value 
of  C.e 

Teak 

745 

818 

9657802 

2462 

15555 

Poon 

579 

596 

6759200 

2221 

14787 

English  Oak 

969 

598 

3494730 

1181 

9836 

Do.  specimen  2 

934 

435 

5806200 

1672 

10853 

Canadian  Oak 

872 

588 

8595864 

1766 

11428 

Dantzic  Oak 

756 

724 

4765750 

1457 

7386 

Adriatic  Oak 

993 

610 

3885700 

1583 

8808 

Ash 

760 

395 

6580750 

2026 

17337 

Beech 

696 

615 

5417266 

1556 

9912 

Elm 

553 

509 

2799347 

1013 

5767 

Pitch  Pine 

660 

588 

4900466 

1632 

10415 

Red  Pine 

657 

605 

7359700 

1341 

10000 

New  England  Fir 

553 

757 

5967400 

1102 

9947 

Riga  Fir 

753 

588 

5314570 

1108 

10707 

Do.  specimen  2 

738 

3962800 

1051 

Mar  Forest  Fir 

696 

588 

2581400 

1144 

9539 

Do.  specimen  2 

693 

403 

3478328 

1262 

10691 

Larch 

531 

411 

2465433 

653 

Do.  specimen  2 

522 

518 

3591133 

832 

Do.  specimen  3 

556 

518 

4210830 

1127 

7655 

Do.  specimen  4 

560 

518 

4210830 

1149 

7352 

Norway  Spar 

577 

648 

5832000 

1474 

12180 

36.  PROBLEMS  IN  DETERMINING  THE  STRENGTH 
OF  TIMBER.* 

1.  To  find  the  strength  of  direct  cohesion  of  a  piece  of 
timber  of  any  given  dimensions. 

Rule. — Multiply  the  number  of  square  inches  in  the  trans 
verse  section  by  the  value  of  C  in  the  table,  (§35,)  and  the 
product  will  be  the  strength  required  in  pounds. 

N.  B.  If  the  specific  gravity  differs  from  the  mean  specific 
gravity  of  the  table,  multiply  the  product  by  the  actual  specific 
gravity,  and  divide  by  the  tabular  specific  gravity  for  the  correct 
strength. 


a  Ingram. 

b  Ultimate  deflection. 


Elasticity.        d  Strength.        e  Cohesion. 


154  STRENGTH   OF   MATERIALS.  [ART.  IX. 

If  i  =  the  breadth  in  inches,  and  rf=the  depth,  iX<*XC  =  W. 
If  either  b  or  d  is  required,  divide  W  by  the  product  of  the  remain 
ing  factors. 

2.  To  find  the  deflection  of  a  beam  fixed  at  one  end,  and 
loaded  with  any  given  weight  at  the  other. 

Rule. — Divide  32  times  the  weight  multiplied  by  the  cube 
of  the  number  of  inches  in  the  length  of  the  beam,  by  the 
continued  product  of  the  tabular  value  of  E,  the  number  of 
inches  in  the  breadth,  and  the  cube  of  the  number  of  inches 
in  the  depth  of  the  beam. 

N.  B.  When  the  beam  is  loaded  uniformly  throughout,  multiply 
the  cube  of  the  length  by  12  times  the  weight,  instead  of  32  times 
the  weight. 

3.  To  find  the  deflection  of  beams  supported  at  both 
ends,  and  loaded  in  the  middle  with  any  given  weight. 

Rule, — Multiply  the  cube  of  the  number  of  inches  in  the 
length,  by  the  number  of  pounds  in  the  given  weight,  and 
divide  by  the  continued  product  of  E,  the  number  of  inches 
in  the  breadth,  and  the  cube  of  the  number  of  inches  in  the 
depth  of  the  beam. 

N.  B.  When  the  beam  is  not  only  supported,  but  is  fixed  at  both 
ends,  the  deflection  is  f  of  that  given  by  the  rule. 

If  the  weight  is  distributed  uniformly  throughout  the  length 
of  the  beam,  the-  deflection  will  be  £  of  that  given  by  the  rule. 

4.  To  find  the  ultimate  deflection  before  their  fracture  of 
beams  or  rods  supported  at  both  ends. 

Rule. — Multiply  U  by  the  number  of  inches  in  the  depth 
of  the  beam,  and  divide  the  square  of  the  number  of  inches 
in  the  length  by  the  result. 

5.  To  find  the  ultimate  transverse  strength  of  any  rec 
tangular  beam  of  timber,  fixed  at  one  end  and  loaded  at 
the  other. 

Rule, — Find  the  continued  product  of  S,  the  number  of 
inches  in  the  breadth,  and  the  square  of  the  number  of 
inches  in  the  depth,  and  divide  the  continued  product  by 
the  number  of  inches  in  the  length. 


§36.]  STRENGTH  OF  TIMBER.  155 

If  W  represents  the  number  of  pounds  that  would  produce 
fracture,  I  the  length  in  inches,  b  the  breadth  in  inches,  and  d  the 
depth  in  inches,  /  =  S  X  *  X  <*2-*-W;  b  =  l  X  W-j-(S  X  ^2);  <*  = 

V  IK  w+T(s~xT)  ;  s  =  /  x  w  -M6  x  <*2)- 

6.  To  find  the  ultimate  transverse  strength  of  any  rec 
tangular  beam  when  supported  at  both  ends,  and  loaded  in 
the  centre. 

Rule.  —  Find  the  continued  product  of  S,  4  times  the  num 
ber  of  inches  in  the  breadth,  and  the  square  of  the  number  of 
inches  in  the  depth,  and  divide  the  product  by  the  number 
of  inches  in  the  length. 

S  X  ^2)  ;    <*  = 


N.  B.  When  the  beam  is  fixed  at  each  end  and  loaded  in  the 
middle,  the  result  obtained  by  the  rule  must  be  increased  one- 
half. 

When  the  beam  is  loaded  uniformly  throughout  its  length,  the 
result  obtained  by  the  rule  must  be  doubled. 

When  the  beam  is  fixed  at  both  ends,  and  loaded  uniformly 
throughout,  the  result  obtained  by  the  rule  must  be  trebled. 

If  the  load  is  to  be  permanent,  it  should  not  exceed  f  of  the 
amount  obtained  by  the  rules. 

7.  To  find  the  weight  under  which  a  given  column  will 
begin  to  bend,  when  placed  vertically  on  a  horizontal  plane. 

Rule.  —  Find  the  continued  product  of  E,  the  cube  of  the 
number  of  inches  in  the  least  thickness,  the  number  of 
inches  in  the  greatest  thickness,  and  .2056.  Divide  this 
continued  product  by  the  square  of  the  number  of  inches  in 
the  length. 

If  W  represents  the  number  of  pounds  that  the  column  can 
sustain,  d  the  number  of  inches  in  the  greatest  thickness,  b 
the  number  of  inches  in  the  least  thickness,  and  I  the  num 
ber  of  inches  in  the  length;  d=  W  X^-r-  (E  X  *3  X  .2056)  ; 
b=ty  W~x  I2  -f-  (.2056  X  E  X  d)  ;  1=  \/E 
E«=W  x  Z2-r-  (d  X  i3  X  -2056). 


156  STRENGTH  OF  MATERIALS.          [ART.  IX. 

37.    EXAMPLES  FOR  THE  PUPIL. 

1.  What  weights  will  be  required  to  tear  asunder  two 
pieces  of  beech,  each  4in.  wide,  and  Sin.  thick,  the  sp.  gr. 
of  the  1st  being  696,  and  that  of  the  2d  678  ? 

2&Ans.  115868  Ib. 

2.  A  red  pine  beam  8Jft.  long,  4in.  broad,  and  6in.  deep, 
is  fixed  at  one  end,  and  loaded  with  a  weight  of  500  Ib. 
Required  the  deflection,  when  the  weight  is  suspended  at 
its  extremity,  and  also  when  it  is  distributed  uniformly 
throughout  the  length  of  the  beam.         1st  Ans.  2.51in. 

3.  A  beam  of  Canadian  oak,  5in.  broad,  Sin.  deep,  and 
25ft.  long,  is  supported  at  both  ends,  and  loaded  with  a 
weight  of  3000  Ib.    Required  the  deflection  when  the  weight 
is  placed  in  the  centre,  and  also  when  it  is  distributed  uni 
formly  throughout  the  length.          2d  Ans.  2.23  inches. 

4.  Required  the  deflection  at  the  instant  of  fracture,  of 
an  ash  beam  30ft.  long,  9in.  wide,  and  6in.  deep. 

Ans.  54.68  inches. 

5.  What  weight  will  be  required  to  break  a  beam  of  larch, 
sp.  gr.  531,  fixed  at  one  end  and  loaded  at  the  other,  the 
breadth  being  3in.,  depth  6in.,  and  length  6ft  ? 

Ans.  979 Jib. 

6.  What  weight  will  be  required  to  break  a  beam  of  pitch 
pine,  supported  at  both  ends  and  loaded  in  the  middle,  the 
length  being  16ft.,  the  breadth  6in.,  and  the  depth  9in.? 
What  weight  would  be  required  if  the  beam  were  fixed  at 
both  ends,  and  loaded  uniformly  throughout  ? 

1st  Ans.  16524  Ib. 

7.  What  weight  will  bend  a  column  of  New  England  fir, 
8ft.  4in.  long,  Sin.  wide,  and  6in.  thick,  placed  vertically 
on  a  plane,  the  weight  being  applied  to  its  upper  extremity  ? 

Ans.  212007.8776  Ib. 

8.  What  weight  would  have  been  required  to  break  the 
beam  mentioned  in  Ex.  6,  if  it  had  been  fixed  at  each  end, 


§38] 


TABLES. 


157 


and  loaded  in  the  middle  ?     If  it  were  merely  supported  at 
each  end,  and  loaded  uniformly  throughout  its  length  ? 

38.  TABLE  SHOWING  THE  NUMBER  OF  POUNDS  THAT 
WILL  PULL  ASUNDER  A  PRISM  ONE  INCH  SQUARE,  OP 
DIFFERENT  MATERIALS,  ACCORDING  TO  THE  EXPERI 
MENTS  OF  M.  MU8CHENBROEK.a 


Cast  sold        . 

.     22000 

2900 

Cast  silver      .     .     . 
Anfle^ea  copper  .     • 

.     41000 
.     34000 

Good  brass  .... 

51000 
16270 

Swedish  copper   . 
Cast  iron    .           .     • 

.     37000 
.     50500 

Horn  

8750 
7500 

Bar  iron,  ordinary   . 
Ditto,  best  Swedish 
Bar  steel,  soft     .     . 
Ditto,  razor  temper 
Cast  tin,  Eng.  block 
Ditto,  grain    .     .     . 
Cast  lead 

.     68000 
.     84000 
.  120000 
.  150000 
.       5200 
.       6500 
860 

COMPOSITIONS. 

Gold  5,  copper  1,  .  . 
Silver  5,  copper  1,  .  . 
Swedish  copper  6,  tin  1, 
Block  tin  3,  lead  1,  .  . 
Tin  4  lead  1  zinc  1,  . 

50000 
48500 
64000 
10200 
13000 

Regulus  of  antimony 
Zinc 

.       1000 
2600 

Lead  8,  zinc  1,  ... 

4500 

ACCORDING  TO  THE  EXPERIMENTS  OF  Mr.  RENNIE. 


—                                                                    ~~~i 

No.  of  Ibs.  that  would 
tear  asunder   a  prism 
1  in.  square. 

Length  in  ft.  that 
would  break  \vith 
its  own  weight. 

Cast  steel        ..... 
Swedish  iron  ..... 
English  iron   ..... 
Cast  iron         ..... 
Cast  copper    ..... 
Yellow  brass  
Cast  tin           .         . 

134256 

72064 
55872 
19096 
19072 
17958 
4736 
1824 

39455 
19740 
16938 
6110 
5092 
5180 
1496 
384 

Good  hemp  rope      .... 
Ditto,  lin.  in  diameter    . 

6400 
5026 

18790 
18790 

39.    THE  LATERAL  STRENGTH  OF  BARS  ONE  FOOT 
LONG,  AND  ONE  INCH  SQUARE.1 


Weight  that  will  bieak  them. 

Weight  they  can  hear 
with  safety. 

Breaking  weight. 

Safe 
weight. 

Tsolb" 

69  Ib. 

Cast  iron  3270  Ib. 
[Oak  627  Ib. 

1090  Ib. 
209  Ib. 

Memel  fir            390  Ib. 
Am.  white  pine  206  Ib. 

Ingram. 


158 


STRENGTH  OF  MATERIALS.  [ART.  IX 


4O.    THE  COHESIVE  FORCE  OF  A  SQUARE  INCH  OF 

IRON    OF    DIFFERENT   KINDS.* 


Iron  wire  .     . 

Dittob    .     . 

Swedish  iron 

Ditto .     . 

Ditto .     . 

Ditto .  . 
English  iron  . 

Ditto .     . 

Ditto  . 


1130771b. 

93964  « 
78850  " 
72064  « 
54960  « 
53244  « 
66000  " 
55000  " 
61600  « 


English  iron 
Welsh  iron . 
Ditto  .  . 
German  iron 
French  iron 
Eussian  iron 
Cast  iron  . 

Ditto    .     . 

Ditto,  Welsh 


65772 Ib. 
64960  « 
55776" 
69133  " 
61000  " 
59472  " 
18295  « 
19488  " 
16255  » 


41.    THE  NUMBER  OF  POUNDS  NECESSARY  TO  CRUSH 
CUBES  OF  1J  INCHES.* 


Aberdeen  granite,  blue  24536 
White-veined  Ital.  marble  21738 
Very  hard  freestone  .  .  21254 
Purbeck  limestone  .  .  .  20610 
Limerick  limestone,  black  19924 
Peterhead  granite  .  .  .  18636 
Compact  limestone  .  .  .  17354 
Yorkshire  paving  stone  .  15856 
Dundee  sandstone  .  ,  .  14919 


Craigleith  stone, 

•with  the  strata  15560 

Ditto  across    ditto  12346 

Cornish  granite 14302 

White  statuary  marble  .  13632 

Fine  brick 3864 

Yellow  baked  brick  .  .  .  2254 

Red  brick 1817 

Pale  red  brick 1265 

Chalk   .  1127 


ONE-INCH  CUBES  WERE  CRUSHED  BY  THE  FOLLOWING 
WEIGHTS  : 

Elm 1284  Ib.  I  English  oak      .     .     .     3860  Ib. 

White  deal  .     .     .     .     1928  Ib.  |  Craigleith  stone    .     .     8688  Ib. 

CUBES    OF   ONE-FOURTH    OF   AN   INCH  WERE    CRUSHED    BY 
THE    FOLLOWING    WEIGHTS  : 


Iron,  cast  vertically  .  11140  Ib. 
Ditto  horizontally  .  10110  " 
Cast  copper 7318  " 


Cast  tin 966  Ib. 

Cast  lead  .  .  483" 


a  Ingram. 

b  The  different  numbers  represent  the  different  results  obtained  by 
the  most  careful  experimenters. 


§43.] 


TABLES. 


159 


TABLE  OF  IRON  AND  HEMPEN  CABLES  or 

EQUAL    STRENGTH.1 


Iron  Cables. 
Diameter  of  Iron  Rod. 

Hemp  Cables. 
Circumference  of  Rope. 

Resistance. 

Inches. 

Inches. 

Tons. 

I 

9 

12 

10 

18 

1* 

11 

26 

l! 

12 

32 

1    5 

13 

35 

if 

14  to  15 

38 

l| 

16 

44 

if 

17 

52 

If 

18 

60 

1} 

20 

70 

2 

22  to  24 

80 

The  stress  given  in  the  above  table  is  the  greatest  to 
which  the  cables  should  be  exposed,  and  is  about  £  the 
breaking  strain. b 

A  "cable's  length,"  is  120  fathoms. 

43.    MEAN  WEIGHT  OF  A  CUBIC  FOOT  OF  STONE,  AND 
THE  WEIGHT  IT  WILL  SUSTAIN  WITH  SAFETY.0 


Weight. 

Pressure. 

Sandy  Bay  granite         .... 
Quincy            "                .... 
Concord          "                .... 
Frankfort       "                 .... 
New  York  white  marble 
N.  Haven  variegated  "           ... 
Penn.  dove  marble         .... 
Vt.         "          "     

168.4811 
167.04 
159 
162 
173 
175 
170 
168 

>. 

197000  Ib. 
156000  « 
149000  ' 
148000  ' 
85000  < 
89000  ' 
86000  < 
86000  ' 

Thomaston  blue  marble 
Connecticut  sandstone   ...» 
North  River         " 
Potomac               "          .... 

179 
164 
156 
153 

90000  « 
118000' 
108000  < 
98000  < 

»Ure. 

b  A  common  rope  1ft.  long,  and  1  inch  in  circumference,  weighs 
.044  to  .046  Ib.  In  a  cable,  it  weighs  .027  Ib.  To  find  the  number  of 
pounds  which  a  rope  will  sustain,  square  the  number  of  inches  in  its  girt, 
and  multiply  by  200  for  commonropes,  and  by  120for  cables. — Tredgold, 

c  Shaw. 


160  STRENGTH   OF   MATERIALS.  [ART.  IX. 

44.  PROBLEMS  ON  THE  STRENGTH  OF  IRON. 

1.  To  find  the  breadth  of  a  uniform  cast-iron  beam,  to 
sustain  a  given  weight  in  the  middle. 

T^Ze.— Multiply  the  number  of  feet  in  the  length,  by  the 
number  of  pounds  to  be  supported,  and  divide  the  product 
by  850  times  the  square  of  the  number  of  inches  in  the 
depth.  The  quotient  will  give  the  number  of  inches  in  the 
breadth. 

When  neither  the  breadth  nor  depth  is  known,  but 
merely  the  proportion  that  exists  between  them;  find  the 
continued  product  of  the  number  of  ft.  in  the  length,  the 
number  of  Ib.  in  the  weight,  and  the  ratio  of  the  depth  to 
the  breadth;  divide  the  continued  product  by  850,  and 
extract  the  cube  root  of  the  quotient.  The  result  will  be 
the  number  of  inches  in  the  depth. 

If  W  =  the  weight  to  be  supported  in  pounds,  I  =  the  length  in 
feet,  b  =  the  breadth  in  inches,  and  d  =  the  depth  in  inches,  W  = 

d§'-4-W. 

2.  To  find  the  proper  breadth  and  depth  of  a  beam  of  cast 
iron  supported  at  both  ends,  when  the  load  is  not  in  the  middle. 

tfufe.— Measure  the  number  of  feet  from  the  point  at 
which  the  weight  is  applied,  to  each  support,  and  find  the 
product  of  the  two  numbers;  divide  4  times  this  product 
by  the  whole  length  between  the  supports,  and  proceed 
with  the  quotient  in  the  same  manner  as  with  the  length 

Problem  1. 

N  B  When  the  load  is  uniformly  distributed  over  the  length 
of  the  beam,  the  depth  need  be  only  |  as  great  as  when  it  is  all 
placed  in  the  middle. 

3.  To  find  the  proper  breadth  and  depth  of  an  iron  beam 
fixed  at  one  end,  and  the  load  applied  to  the  other,  or  of  a 
beam  supported  upon  a  centre  of  motion. 

Rule  —Take  the  length  from  the  point  at  which  the  beam 
is  fixed,  or  from  the  centre  of  motion,  to  the  point  where  the 


§44.]        PROBLEMS   ON    THE   STRENGTH   OP   IRON.  161 

load  is  applied,  and  calculate  the  strength  by  the  rules 
in  Problem  1,  using  instead  of  850,  212  for  cast  iron,  238 
for  wrought  iron,  or  425  when  the  weight  is  uniformly  dis 
tributed  over  the  length  of  the  beam. 

4.  To  find  the  proper  depth  of  the  teeth  of  wheels. 
Rule. — Divide  the  number  of  pounds  which  represents 

the  stress  of  the  wheel  at  the  pitch-circle,*  by  1500,  and 
extract  the  square  root  of  the  quotient.  The  result  will  give 
the  thickness  of  the  teeth  in  inches. 

N.  B.  The  length  of  the  teeth  ought  not  to  exceed  their  thick 
ness.  The  breadth  should  be  in  proportion  to  the  stress  upon 
them,  and  this  stress  should  not  exceed  400  Ib.  for  each  inch  in 
breadth. 

5.  To  find  the  proper  thickness  of  the  teeth  of  a  wheel, 
when  the  power  of  the  first  mover  is  given  in  pounds,  and 
the  velocity  per  second  in  feet. 

Rule. — Form  the  continued  product  of  the  numbers  which 
represent  the  power  and  velocity  per  second  of  the  first  mover, 
and  .073;  multiply  the  number  of  revolutions  the  wheel  is 
to  make  per  minute,  by  the  radius  the  wheel  should  have  if 
its  pitch  were  two  inches ;  divide  the  first  of  these  products 
by  the  second,  and  the  cube  root  of  the  quotient  will  give 
the  thickness  of  the  teeth  in  inches. 

6.  To  find  the  proper  diameter  of  a  solid  cylinder  of  cast 
iron  to  sustain  a  given  weight,  when  supported  at  both  ends, 
and  the  weight  applied  at  the  middle  of  the  length. 

Rule. — Take  the  weight  in  pounds,  and  the  length  in  feet; 
multiply  the  two  numbers  together,  and  divide  the  product 
by  500 ;  the  cube  root  of  the  quotient  is  the  number  of  inches 
in  the  diameter. 

If  W  represents  the  weight  in  pounds,  I  the  length  in  feet,  and 
rfthe  diameter  in  inches,  W  =  500  X  e?3-M;  2  =  500  X  ^3-r-W 

a  The  pitch  of  the  teeth  of  a  wheel,  is  the  distance  between  ths 
middle  points  in  the  bases  of  two  adjacent  teeth.     It  should  be  at  least 
2.1  times  the  thickness  of  the  teeth. 
11 


162  STRENGTH   OF   MATERIALS.  [ART.  IX. 

7.  To  find  the  proper  diameter  of  a  solid  cylinder  of  cast 
iron  supported  at  both  ends,  to  bear  a  given  weight  when 
the  strain  is  not  in  the  middle. 

Rule.  —  Take  the  weight  in  pounds,  and  the  distances  in 
feet,  from  the  point  where  the  weight  is  applied  to  each  of 
the  points  of  support;  divide  the  continued  product  of  these 
three  numbers  by  the  number  of  feet  in  the  distance  between 
the  points  of  support,  and  cut  oif  three  figures  from  the  right 
hand.  The  cube  root  of  the  result  will  give  one  half  of  the 
diameter  of  the  cylinder  in  inches. 

8.  To  find  the  proper  diameter  of  a  solid  cylinder  of  cast 
iron  when  supported  at  both  ends,  to  sustain  a  load  uniformly 
distributed  over  its  length. 

Rule.  —  Multiply  the  number  of  feet  in  the  length  by  the 
number  of  pounds  in  the  weight,  and  y1^  of  the  cube  root 
of  the  product  will  be  the  number  of  inches  in  the  diameter. 

W  =  1000x  dz-+-l;  Z=aOOOx  <*3-~W. 

9.  To  find  the  proper  length  of  a  solid  cylinder  of  cast  iron, 
when  fixed  at  one  end  and  loaded  at  the  other;  also  when 
the  cylinder  is  supported  on  a  centre  of  motion. 

Rule.  —  Take  the  weight  in  pounds,  and  the  distance  of 
the  weight  from  the  point  of  support  in  feet,  multiply  the 
two  numbers  together,  and  J  of  the  cube  root  of  the  product 
will  be  the  number  of  inches  in  the  diameter. 


10.  The  strength  of  direct  cohesion,  of  the  materials  in 
tables,  §38  and  40,  may  be  found  by  Problem  1,  §36, 
using  the  numbers  in  those  tables  opposite  to  the  given 
material,  instead  of  the  value  of  C. 

11.  The  lateral  strength  of  iron  may  be  found  by  the  rules 
for  that  of  timber,  using  the  number  opposite  to  iron  in  §  39, 
instead  of  the  value  of  S  in  §  35. 

12.  The  strength  of  a  column  to  resist  being  crushed,  is 
proportioned  to  the  area  of  its  transverse  section.     Hence, 


§45.]  MISCELLANEOUS   EXAMPLES.  163 

to  find  the  weight  which  will  crush  any  column,  multiply 
the  number  of  inches  in  the  area  of  its  transverse  section 
by  the  proper  number  in  §  41,  and  divide  the  product  by  2 J; 
or  multiply  the  number  of  feet  in  the  area  of  the  transverse 
section,  by  the  pressure  given  in  §  43.  The  area  of  a  cylin 
drical  column  may  be  estimated,  in  making  this  calculation, 
at  |  of  the  square  of  the  diameter.  The  true  area  of  a  circle, 
is  found  by  multiplying  the  square  of  the  diameter  by  .7854. 
The  result  is  the  number  of  pounds. 

45.    EXAMPLES  FOR  THE  PUPIL. 

1.  What  is  the  breadth  of  a  cast-iron  beam  30ft.  long, 
and  Sin.  deep,  that  will  support  a  weight  of  8  tons,  placed 
in  the  middle  ?     If  the  length  is  30ft.  and  the  breadth  6in., 
what  must  be  the  depth  to  support  10  tons  ? 

1st  Am.  9.88in. 
2d    «    11.47" 

2.  The  front  of  a  house  is  to  be  broken  out  to  make  shops, 
and  the  front  wall,  which  is  40ft.  long,  is  to  be  supported 
by  2  cast-iron  beams,  with  a  prop  under  the  middle  of  the 
wall.     If  there  are  4000  c.  ft.   of  wall  to  be  supported, 
weighing  140  Ib.  per  c.  ft.,  what  must  be  the  breadth  and 
depth  of  each  beam,  the  depth  being  5  times  the  breadth  ? 

Am.  Depth  20.352in. ;  Breadth  4.0704in. 

3.  The  second  story  of  a  building  is  to  project  3ft.  over 
the  first.     What  must  be  the  depth  of  the  fixed  iron  beams, 
which  are  4  inches  broad,  supposing  the  weight  supported 
by  each  to  be  33600  Ib.  ?  Ans.  7.7in. 

4.  If  the  greatest  stress  at  the  pitch-circle  of  a  wheel  is 
6500  Ib.,  what  should  be  the  thickness  of  the  teeth  ? 

Ans.  2.08in. 

5.  If  the  effective  force  of  the  piston  of  a  steam  engine 
is  10000  Ib.,  and  its  velocity  5  ft.  per  second,  what  should 


164  STRENGTH   OF   MATERIALS.  [ART.  IX. 

be  the  thickness  for  the  teeth  of  a  wheel,  which  is  to  make 
20  revolutions  in  a  minute,  and  to  have  120  teeth  ? a 

Ans.  1.46in. 

6.  What  weight  will  a  cast-iron  cylinder,  supported  it 
both  ends,  sustain  in  the  middle  of  its  length,  the  diameter 
being  8  inches,  and  the  length  16£  ft.  ?     Ans.  15515  Ib. 

7.  What  must  be  the  diameter  of  a  cast-iron  cylinder, 
which  is  20ft.  long,  to  sustain  a  weight  of  33600  Ib.,  the 
weight  being  applied  at  the  distance  of  5ft.  from  one  end  ? 

Ans.  10.026in. 

8.  A  load  of  25000  Ib.  is  to  be  uniformly  distributed  over 
a   solid  cast-iron  cylinder.     Required  the  length  of  the 
cylinder,  the  diameter  being  9  inches.          Ans.  29.16ft. 

9.  A  solid  cylinder  of  cast  iron,  8  inches  in  diameter,  is 
supported  in  the  middle.     What  may  be  the  length  of  the 
arms,  to  support  10000  Ib.  at  the  extremity  of  each  ? 

Ans.  6.4ft. 

10.  What  weight  would  pull  asunder  a  hemp  rope  2  £  inches 
in  diameter?  Ans.  31412 Jib. 

11.  What  weight  distributed  uniformly  over  a  cast-iron 
beam,  20ft.  long,  6in.  broad,  and  Sin.  deep,  will  break  the 
beam,  it  being  supported  at  both  ends  ?     Ans.  41856  Ib. 

12.  What  weight  can  be  sustained  with  safety  by  each 
of  the  marble  pillars  of  Girard  College,  estimating  their 
strength  as  equivalent  to  that  of  Pennsylvania  Dove  marble, 
the  least  diameter  of  the  pillars  being  5ft  ? b 

Ans.  1688610  Ib.,  or,  estimating  the  area  at  |  of  the 
square  of  the  diameter,  1672222  Ib. 

13.  The  length  of  each  arm  of  the  wrought-iron  beam 
of  a  balance  is  3ft.,  and  the  depth  is  8  times  the  breadth. 

*  The  circumference  of  a  wheel  with  120  teeth,  and  a  pitch  of  2in., 
is  120  X  2  =  240in.  The  radius  of  such  a  wheel  would  be  240  -7-  (2  X 
3.1416)  =  38.197in.  See  the  article  on  Mensuration. 

"Architect's  Report. 


§46.]  TABLE  OP  SPECIFIC  GRAVITIES.  165 

Required  the  depth  and  breadth  necessary  to  enable  the 
balance  to  weigh  &  a  ton. 

Ans.  Depth,  V8  X  3  X  1120-f-238=4.834in. 
Breadth,  .604in. 

14.  What  should  be  the  depth  and  breadth  of  a  cast-iron 
beam,  30ft.  long,  to  support  a  weight  of  20  tons,  placed 
10ft.  from  one  end,  the  beam  being  supported  at  both  ends, 
and  the  depth  being  4  times  the  breadth  ? 

Ans.  Depth,  17.7812in. ;  Breadth,  4.4453in. 


X.    SPECIFIC  GRAVITY. 

THE  Specific  Gravity  of  a  body,  is  the  ratio  of  its  weight 
to  the  weight  of  an  equal  volume  of  some  other  body  assumed 
as  a  standard.  The  standard  usually  adopted  for  this  pur 
pose  is  pure  distilled  water  at  a  given  temperature.  In 
England,  the  temperature  of  62°  Fahrenheit  is  generally 
taken ;  the  French  take  32°,  or  the  temperature  of  melting 
ice.a 

46.    TABLE  OF  SPECIFIC  GRAVITIES. 
Compiled  from  the  Encyclopedia  Britannica  and  other 


ACACIA,  inspissated  juice  1513 

Acid,  acetic  1007  to  1009.5 

acetous  1009.5  to  1025.1 

arsenic  3391 

boracic,  scales         1475 

citric  1034.5 

fluoric  1500 


Acid,  muriatic  1284.7 

nitric       1271.5  to  1583 
phosphoric,  liquid  1417 
solid    2852 

sulphuric  1840. 9  to  2125 
Agate  2348  to  2666.7 

Air  1.2308 


a  Brande. 

b  The  specific  gravity  of  water  is  fixed  at  1000.  As  a  cubic  foot  of 
water  weighs  1000  ounces  avoirdupois,  the  specific  gravity  of  each 
article  named  in  the  table  will  represent  the  weight  of  1  cubic  foot. 


166 


SPECIFIC   GRAVITY. 


[ART. 


Alabaster  2611  to  2876.1 

Alcohol,  absolute  791 

mixed  with 

water  829.3  to  991.9 
Alder  wood  800 

Alum  1750 

Amber  1078  to  1085.5 

Antimony,  fused  6624  to  6860 
Apple  tree  793 

Arsenic,  fused  8310 

glass  of,  (arsenic 

of  the  shops)     3594.2 
Asbestos  2577.9  to  3080.8 

Do.  mountain  cork  680.6  to  993. 3 
Ash  727a  to  845 

Asphaltum  1070  to  2060 

BASALT  2421  to  3000 

Beech  6961  to  852 

Beryl,  oriental  3549.1 

aquamarine  2650  to  2759 
Bismuth,  molten  9756  to  9822 
Blood,  human  1054 

Bone  of  an  ox  1656 

Borax  1740 

Boxwood  912  to  1328 

Brass,  common    7824  to  8395b 

wiredrawn  8544 

Brazil  wood  1031 

Brick  1557a  to  2000 

Brickwork6  1872 

Butter  942.3 

CADMIUM  8604  to  8694.4 

Camphor  988.7 

Cannel  coal  1270 

Caoutchouc  933.5 

Castor  oild  970 

Cedar  457*  to  561 


Chalk8 
Cherry 
Chestnut1 
Chromium"1 


2252  to  2657 

715 

610 

5900 


Citron  wood  726.3 

Claya  2000 

Coal,  bituminous'  1262  to  1364 
anthracitef  1500 

Cobalt,  fused       7645  to  7811 
Cocoa  wood  1040.3 

Cokea  744 

Copal  1045.2  to  1139.8 

Copper,?  native    7600  to  8508.4 
fused     7788  to  8607 
wiredrawn          8878 
Coralh  2680 

Cork  240 

Cypress  644 

DIAMOND  3444.4  to  3550 

EARTH,  common1 1520  to  2000 
mean  density'        5670 
Ebony  1209  to  1331 

Elm  671 

Emerald  2600  to  3155.5 

Emeryh  4000 

Ether,  acetic  866.4 

muriatic  729.6 

nitric  908.8 

sulphuric      716  to  745 
FAT  923.2  to  936.8 

Felspar  2438  to  2704 

Filbert  tree  600 

Fir  498  to  553 

Flint  2243.1  to  2664.4 

GAKNET,  common  3576  to  3688 
precious  4085  to  4352 
Gas,  ammonia  .73459 


*  Cavallo.  b  369  c.  in.  =  Icwt.          c  Benjamin. 

d  Ingram.  e  13  c.  ft.  =  1  ton.  f  W.  R.  Johnson. 

f  A  square  foot  of  sheet  copper,  £in.  thick,  weighs  11  Ib.  12oz. 
h  Barlow.  *  Cavendish. 


§46.] 


TABLE   OF    SPECIFIC   GRAVITIES. 


167 


Gas, 

atmospheric  air 

1.2308 

Gum  ammoniac 

1207.1 

carbonic  acid 

1.87 

Arabic 

1452.3 

carbonic  oxide 

1.1777 

guaiacum 

1228.9 

carburetted  hydro.  .73848 

lac 

1139 

chlorine 

3.0401 

tragacanth 

1316.1 

cyanogen 

2.2228 

Gunpowder,  solid 

1745 

fluosilicic  acid 

4.3984 

shaken 

932 

hydriodic  acid 

5.4684 

Gypsum 

1872 

to  3310.8 

hydrogen 

.0898484 

HAZEL 

606 

muriatic  acid 

1.5353 

Hone 

2876.3 

to  3127.1 

nitrogen 

1.928 

Honey 

1450 

nitrous 

1.2786 

Hornblende 

3333 

to  3830 

nitrous  acid 

3.908 

ICEb 

930 

nitrous  oxide 

1.9865 

Indigo 

769 

olefiant 

1.20377 

Iodine 

4948 

oxygen 

1.3588 

Iridium,  fused 

18680 

phosph.  hydrogen  1.0708 

Iron,0  bar 

7600 

to  7788 

steam 

.76739 

cast 

6953 

to  7295 

sulph.  hydrogen 

1.4661 

magnetic 

4518 

sulphurous  acid 

2.6097 

meteoric 

6480 

Glass,  bottle 

2732.5 

Ivory 

1825 

to  1917b 

crown          2487 

to  2520 

JARGON,  of  Ceylon'1 

4416 

flint             3000 

to  3437 

Jasper 

2358.7 

to  2816 

green 

2642.3 

Jet 

1259 

plate           2520 

to  2760 

Juniper  tree 

556 

Gold 

,  not  hammered 

19258^.7 

LARD 

947.8 

hammered 

19342 

Lead 

11352  to  11445 

Am.  standard 

17350* 

Lignum  Vitaa 

1333 

Eng.       « 

18888 

Limestone 

1386.4 

to  3183 

French  " 

17486 

Linden 

604 

Granite                 2613 

to  2760.9 

Linseed  oil 

940.3 

a  The  specific  gravity  of  the  Mint  standard  gold  varies  from  17250  to 
17500,  according  to  the  greater  or  less  proportion  of  copper  used  in  the 
alloy.  The  average  specific  gravity  is  about  17350. 

b  Barlow. 

c  A  square  foot  of  cast  iron,  ?in.  thick,  weighs  9lb.  10.6oz ;  a  sq.  ft. 
of  malleable  iron  of  the  same  thickness,  9  Ib.  15.2  oz. ;  a  bar  1ft.  long 
and  Hin.  square,  of  cast  iron,  9lb.  8oz. ;  a  bar  of  malleable  iron,  of 
the  same  size,  9lb.  lUoz. ;  a  round  iron  rod,  1ft.  long  and  Uin.  in 
diameter,  7lb.  9.2  oz. 

d  Ingram. 


168 


SPECIFIC   GRAVITY. 


[ART.  X. 


Living  men1 
Loadstone 
Logwood 
MADDER 


891 

4200  to  4900 
913 
765 


Magnesia,  sulphate  of 
Mahogany 


1797.6 
1063 
6850 

Maple  755 

Marble,  Carrara  2716 

Egyptian  2668 

various    2516  to  2858 
Mercury  13568 

Mica  2883 

Milk  1020.3  to  1040.9 

Molybdena,  native  4738.5 

Mortar,  dryb        1384  to  1893 
Muriatic  acid  1284.7 

NAPHTHA  847.5 

Nickel,  metallic  7421  to  9333.3 

forged  8600 

Nitre  1900  to  2246 

Nitric  acid,  (aquafortis)  1500* 

Nitrous  acid  1452a 

OAKb  748  to    993 

heart  of  1170 

Obsidian  2348 

Oil,  of  turpentine  870 

olive  915.3 

whale  923.3 

various         857.7  to  1044 

Opal  1958  to  2600 

Opium  1336.5 

Orange  tree  705.9 

PALLADIUM  11800 

Pear  tree  661 

Pearls  2683 

Peat  600  to  1329 

Pewter*  7471 

Phosphorus  1770 


Pine  540  to  683 

Pitch*  1150 

Platina  14626  to  22069 

Plumbago  1987  to  2267 

Poplar  360  to  529.4 

Porcelain,  China  2384.7 

do.  European    2145  to  2545 
Potash,  carbonate  1459.4 

Potassium  972.23 

Proof  spirit  916 

Pumice  stone  914.5 

QUARTZ  2652 

Quince  tree  705 

RHODIUM  11000 

Rock  crystal      2581  to  2888 
Rosina  1100 

Ruby  3531  to  4283.3 

SANDb  1454  to  1886 

Sandstone          2142  to  2483.5 
Sapphire  3130  to  4283 

Scythe  stone,  fine  2609 

Serpentine        2264  to    3000 
Silver  10474  to  10610 

Slate  26718  to   2752b 

Soda,  sulphate  1439.8 

carbonate  1000  to  1500 
Sodium  865.07 

Spar,  heavy*  4430 

Spermaceti  943.3 

Steel  7767  to  7840.4 

Stone,  common  2000  to  2700 
rotten  1981 

Sugar,  white  1606 

Sulphur,  native  2033.2 

Sulphuric  acid  1841 

TALLOW  941.9 

Tar*  1015 

Tellurium,  native  5700  to  6100 
Tin  7063  to  8487 


*  Ingram. 


Cavallo. 


§47.] 


PROBLEMS  IN   SPECIFIC  GRAVITY. 


169 


Topaz 

Tourmaline 

Tungsten 

Turpentine 

spirits  of 

ULTRAMARINE 

Uranium 

VAPOR  of  alcohol 

do.         absolute 
hydriodic  ether 
muriatic  ether 
sulphuric  ether 
iodine 


3531  to  4061.5 
3086  to  3362 
4355  to  6066 
991 
870 
2360 
7500 
2.58468 
1.985 
6.7884 
2.731 
3.182 
10.6089 


oil  turpentine        6.17 
sulph.  of  carbon  3.255 
water  .76739 

Vinegar  1013.5  to  1080 

WALNUT  671 

Water,  distilled  1000 

Dead  Sea  1240.3 


Water,  sea  1026.3 

well  1001.7 

Wax,  bees'  964.8 

shoemakers'  897 

Whalebone  1300 

Willow  585 

Wine,  Burgundy  991.5 

Canary  1033 

.    Champagne  997.9 

Malaga  1022.1 

Malmsey  1038.2 

Port  997 

Tokay  1053.8 

Wolfram                5705  to  7333 

Wootz,  hammered*  7787 

YEW,  Dutch  788 

Spanish  807 

ZINC,  common  6862 
pure  &  compressed  7190.8 


47.    PROBLEMS  IN  SPECIFIC  GRAVITY.* 

I.  To  find  the,  magnitude  of  a  body  from  its  weight. 
Find  the  weight  of  the  body  in  ounces,  and  divide  by  the 

specific  gravity.     The  quotient  will  be  the  number  of  cubic 
feet  in  the  contents. 

II.  To  find  the  weight  of  a  body  from  its  magnitude. 
Find  the  number  of  cubic  feet  in  the  body,  and  multiply 

by  the  specific  gravity.     The  product  will  be  the  number 
of  ounces  in  the  weight. 

III.  To  find  the  specific  gravity  of  a  body. 
CASE  I.  When  the  body  is  heavier  than  water. 
Weigh  the  body  both  in  air  and  in  water.     Annex  three 


Ingram. 


170  SPECIFIC    GRAVITY.  [ART.  X. 

ciphers  to  the  weight  in  air,  and  divide  by  the  difference  of 
the  weights.     The  quotient  will  be  the  specific  gravity. 

CASE  II.  When  the  body  is  lighter  than  water. 

Having  weighed  the  light  body  in  air,  and  a  body  heavier 
than  water  both  in  air  and  water,  fasten  them  together  with 
a  slender  tie,  then  weigh  the  compound  in  water,  and  sub 
tract  its  weight  from  the  weight  of  the  heavy  body  in  water; 
to  the  remainder  add  the  weight  of  the  light  body  in  air, 
and  by  the  sum  divide  one  thousand  times  the  weight  of  the 
light  body  in  air.  The  quotient  will  be  the  specific  gravity 
of  the  light  body. 

IV.  To  find  the  quantity  of  each  ingredient  in  a  mixture 
of  two  substances* 

1.  Multiply  the  specific  gravity  of  the  mass  by  the  dif 
ference  between  the  specific  gravities  of  the  two  ingredients, 
for  a  first  product. 

2.  Multiply  the  specific  gravity  of  that  ingredient  whose 
quantity  is  desired,  by  the  difference  between  the  specific 
gravity  of  the  mass,  and  that  of  the  other  ingredient,  for  a 
second  product. 

3.  Multiply  the  whole  weight  of  the  mass  by  the  second 
product,  and  divide  by  the  first  product.     The  quotient  will 
be  the  weight  of  the  ingredient  sought. 

48.    EXAMPLES  FOR  THE  PUPIL. 

1.  How  many  cubic  inches  in  1  Ib.  of  white  sugar  ? 

Ans.  sfgC.  ft. =17.215  c.  in. 

2.  A  keg  is  found  to  contain  13790  cubic  inches.     What 
weight  of  butter  will  it  hold?  Ans.  470 Ib. 

3.  A  piece  of  Quincy  granite  weighs  25  Ib.  12ioz.  in  air, 
and  16 Ib.  IJoz.  in  water.     What  is  its  specific  gravity? 

Ans.  2661. 

a  The  student  will  observe  that  this  is  a  case  in  Alligation. 


§48.]  MISCELLANEOUS   EXAMPLES.  171 

4.  A  piece  of  copper  weighs  18  Ib.  in  air  and  16  Ib.  in 
water.     A  piece  tif  elm,  which  weighs  15  Ib.  in  air,  is  fas 
tened  to  the  copper,  and  the  compound  weighs  6  Ib.  in  water. 
What  is  the  specific  gravity  of  the  elm  ?  Ans.  600. 

5.  What  quantity  of  gold,  sp.  gr.  19258,  and  of  silver, 
sp.  gr.  10474,  must  be  mixed  to  form  a  mass  weighing  Icwt. 
3qr.  4  Ib.,  and  having  a  specific  gravity  of  16000  ? 

Ans.  151.44  Ib.  gold,  48.56  Ib.  silver. 

6.  What  are  the  cubical  contents  of  a  pillar  of  Pennsyl 
vania  marble,  sp.  gr.   2720,  the  weight  being  63T.  8cwt. 
21  Ib.?  Ans.  835  c.  ft.  884  c.  in. 

7.  Each  of  the  pillars  of  Girard  College  is  55ft.  high, 
and  6ft.  in  diameter  at  the  base.     What  would  be  the  weight 
of  a  square  block  of  marble  from  which  a  column  of  the 
same  size  could  be  cut,  the  specific  gravity  being  2716  ? 

Ans.  150T.  3qr.  21  Ib. 

8.  It  is  proposed  to  float  500  c.  ft.  of  granite  on  a  pine 
raft  50  ft.  long,  and  20ft.  wide.     What  must  be  the  depth 
of  the  raft  in  order  that  it  may  float  at  least  6  inches  above 
the  water,  the  specific  gravity  of  the  granite  being  2620, 
and  the  sp.  gr.  of  the  pine  600  ?a 

9.  A  raft  of  elm  is  3ft.  6in.  thick.     To  what  depth  will 
it  sink?  Ans.  2ft.  4.182in. 

10.  What  must  be  the  depth  of  a  cedar  raft,  16ft.  long 
and  10ft.  wide,  to  float  10000  Ib.  of  bricks,  the  cedar  being 


a  All  floating  bodies  sink  till  they  have  displaced  a  quantity  of 
fluid  equivalent  to  their  own  weight.  In  this  example,  the  granite 
would  cause  the  raft  to  displace  1310  c.  ft.  of  water,  to  do  which,  it 
must  sink  1310 ~- (50  X  20)=  1.31ft.  to  which  the  Gin.  =  . 5ft.  should 
i^e  added,  making  1.81ft.  to  represent  the  buoyancy  of  the  pine.  As 
•he  sp.  gr.  of  the  pine  is  |  that  of  water,  it  will  sink  |  of  its  depth, 
leaving  |  for  buoyant  force.  1.81ft.  must  therefore  be  g  of  the  dspth 
of  the  raft,  and  the  depth  must  be  1.81-f- 1  =  4.525ft. 


172  SPECIFIC   GRAVITY.  [ART.  X. 

of  the   sp.   gr.   550,   and   the   raft  floating   Sin.   out   of 
water  ?  Ans.  2ft.  9^in. 

11.  What  weight  will  a  raft  30ft.  long,  16ft.  wide,  and 
3ft.  deep  sustain,  and  float  6  inches  above  water,  the  specific 
gravity  of  the  raft  being  580  ?  Ans.  22800  Ib. 

N.  B.  First  find  how  much  above  water  the  raft  would  float  if  it 
were  not  loaded,  and  subtract  6  inches  to  find  how  much  it  is  sunk 
by  the  load.  The  weight  of  the  load  will  then  be  equivalent  to  the 
weight  of  the  quantity  of  water  displaced  by  it. 

12.  How  many  inches  above  water  will  a  raft  float,  if 
loaded  with  8500  Ib.,  the  raft  being  12ft.  wide,  20ft.  long, 
and  2ft.  deep,  and  of  the  specific  gravity  of  625  ? 

Ans.  2.2  inches. 

N.  B.  First  find  the  entire  weight  of  the  raft  and  load,  and  see 
what  depth  of  water  must  be  displaced  to  yield  the  same  weight. 
Subtract  the  depth  of  water  displaced, from  the  depth  of  the  raft, 
and  the  remainder  will  give  the  part  out  of  water. 

13.  What  is  the  weight  of  a  sheet  of  malleable  iron,  3ft. 
6in.  wide,  8ft.  long,  and  y^in.  thick? 

Ans.   69  Ib.  10.4oz. 

14.  What  is  the  weight  of  an  iron  rod,  3in.  in  diameter, 
and  16£ft.  long?a     Of  a  rod  lin.  in  diameter,  and  lOJft. 
long  ?  Ans.  4cwt.  Iqr.  23  Ib.  15.2oz. ; 

Iqr.  61b.  S.loz. 

15.  What  is  the  weight  of  a  sheet  of  copper,  3ft.  wide, 
6ft.  Sin.  long,  and  Jin.  thick  ?          Ans.  Icwt.  5  Ib.  Soz. 

16.  The  amount  of  water  displaced  by  a  loaded  ship,  is 
found  to  be  96000  cubic  feet.     Required  the  weight  of  the 
vessel  and  cargo,  the  water  being  of  the  specific  gravity  of 
1020.  Ans.  2732T.  2cwt.  3qr.  12  Ib. 


•  The  weight  of  iron  rods  of  the  same  length,  is  proportioned  to  tht> 
squares  of  their  diameters. 


§49.]  GENERAL  REMARKS.  173 

XL    THE  EOAD.a 
49.    GENERAL  KEMARKS. 

WHEN  it  is  proposed  to  construct  a  road,  the  engineer 
first  makes  himself  acquainted  with  the  face  of  the  country 
through  which  the  road  is  to  pass,  and  selects  what  he  con 
siders  as  the  best  general  route.  An  instrumental  survey 
is  then  made  of  the  country  along  the  proposed  route,  taking 
levels  from  point  to  point,  throughout  the  whole  distance, 
to  determine  the  requisite  inclinations  of  the  slopes  of  the 
cuttings  and  embankments,  and  making  borings  in  all  places 
where  excavations  are  required,  to  determine  the  strata 
through  which  the  cuttings  are  to  be  made.  A  plan  and 
section  are  then  drawn,  exhibiting  the  results  of  this  inves 
tigation. 

In  selecting  the  route,  regard  should  be  had  to  the  supply 
of  materials  for  constructing  the  road,  and  for  keeping  it 
in  repair.  Therefore  the  position  of  gravel  pits  and  quarries 
in  the  neighborhood  of  the  proposed  line,  should  be  well 
ascertained. 

The  expense  of  construction  should  be  proportioned  to  the 
traffic  expected  on  the  road.  If  the  amount  of  travel  will 
be  great,  all  steep  acclivities  should  be  avoided,  either  by 
cutting  down  the  hills  and  filling  up  the  valleys,  or  by  pass 
ing  around  the  base  of  the  hills. 

It  is  recommended  by  some  writers  to  avoid  a  dead  level, 
as  a  moderate  inclination  of  the  surface  facilitates  drainage, 
and  tends  to  keep  the  road  dry.  But,  if  proper  attention 
is  paid  to  the  form  of  the  road,  there  will  be  no  difficulty 
in  keeping  it  properly  drained,  without  resorting  to  any 
expedient  that  will  be  necessarily  attended  with  a  loss  of 
power. 

*  Brande,  Gillespie,  Mahan. 


174  THE   ROAD.  [ART.  XI. 

The  top  should  be  slightly  rounded,  being  made  highest 
in  the  middle,  and  gradually  sloping  to  a  trench  at  each  side, 
so  that  all  the  water  may  be  carried  off.  The  surface  should 
be  as  hard  and  smooth  as  possible,  and,  whenever  repairs  are 
required,  broken  stone,  pebbles,  or  hard  gravel  should  be 
used,  if  it  is  possible  to  obtain  either  of  them. 

«5O.    EXAMPLES  FOR  THE  PUPIL. 

1.  How  many  acres  per  mile  will  be  taken  up  by  a  road 
that  is  2  rods  wide  ? — by  a  road  40ft.  wide  ? — by  a  road 
60ft.  wide? 

2.  In  1678,  a  contract  was  made  to  establish  a  coach 
between  Edinburgh  and  Glasgow,  a  distance  of  44  miles. 
The  coach  was  to  be  drawn  by  6  horses,  and  the  journey 
between  the  places,  to  and  from,  was  engaged  to  be  completed 
in  6  days.a     At  what  average  rate  did  the  coach  move,  if  it 
travelled  9  hours  per  day? 

3.  In  the  year  1763,  there  was  but  one  line  of  stage-coaches 
between  Edinburgh  and  London,  which  started  once  a  month 
from  each  place.     It  then  took  a  fortnight  to  perform  the 
journey,  which  is  now  completed  in  less  than  48  hours.1    The 
number  of  persons  travelling  between  the  two  places  did  not 
probably  exceed  50  per  month,  but  the  present  intercourse 
is  supposed  to  amount  to  at  least  300  per  day.     What  has 
been  the  rate  of  increase,  both  in  the  rate  of  travel  and  in 
the  number  of  passengers  ? 

4.  What  will  be  the  cost  of  a  plank  road  per  mile,  for  a 
single  track  8ft.  wide  and  3in.  thick,  with  two  sills,  each 
4in.  square,  at  $4.50  per  M.,b  the  laying  and  grading  being 
75cts.  per  rod,  and  superintendence  $75  per  mile  ? 

*  Brande.  Even  so  recently  as  the  year  1750,  the  stage-coach  from 
Edinburgh  to  Glasgow  took  a  day  and  a  half  to  make  the  journey. 

b  "  Wood  is  paid  for  by  the  cubic  foot,  unless  some  one  of  its  dimen 
sions  is  as  small  as  4  inches,  when  board  measure  is  used." — Gillespie. 


§50.]  MISCELLANEOUS   EXAMPLES.  175 

5.  Wishing  to  know  my  distance  from  the  foot  of  a  steeple 
which  is  125ft.  high,  I  hold  a  foot  rule  at  arm's  length, 
(which  I  have  found  to  be  equivalent  to  2ft.  Sin.  from  the 
eye,)  and  find  that  If  inches  on  the  rule  intercepts  the  rays 
from  the  top  and  base  of  the  steeple.    What  is  the  distance  ? a 

Am.  19284ft. 

6.  The  mean  velocity  of  sound  through  the  atmosphere  is 
about  1090ft.  per  second. b    What  is  the  distance  of  a  hill  on 
which  a  cannon  is  fired,  if  8J  seconds  elapse  between  the 
flash  and  report  ? 

7.  If  two  supports  of  a  rail  that  are  3ft.  apart,  vary  £  of 
an  inch  from  an  exact  level,  to  what  elevation  per  mile  would 
the  ascent  be  equivalent  ? 

8.  Estimating  the  cost  of  a  railroad  at  $30000  per  mile, 
and  the  annual  repairs  and  expenses  at  $2000  per  mile,  how 
much  might  be  profitably  expended  to  shorten  the  road  1 
mile,  the  rate  of  interest  being  6  per  cent.  ? c — to  shorten  it 
2m.  6fur.  23r.  ?  2d  Am.  $178718.75. 

9.  An  embankment  of  27000  c.  ft.  is  to  be  made.     It  is 
estimated  that  1  man  can  loosen  20  c.  yd.  per  day,  or  load 
in  barrows  25  c.  yd.,  or  transport  30  c.  yd.,  or  spread  and 
level  80  c.  yd.     According  to  this  estimate,  what  will  be 
the  cost  of  the  whole,  allowing  $1.25  per  day  for  wages, 
and  10  per  cent,  for  shrinkage  of  the  earth,  and  adding  ^5 
of  the  amount  of  wages  for  tools  and  superintendence,  and 

a  The  length  marked  on  the  rule,  is  to  the  distance  of  the  rule 
from  the  eye,  as  the  height  of  the  object  is  to  its  distance.  Distances 
may  also  be  conveniently  measured  by  pacing,  or  by  noting  the  time 
that  elapses  between  the  flash  and  report  of  a  gun. 

b  Pierce. 

c  In  laying  out  a  road  of  any  kind,  the  preference  never  should  be 
given  to  the  longer  of  two  proposed  routes,  merely  because  it  can  be 
constructed  at  a  less  expense.  In  order  to  shorten  the  distance,  the 
difference  in  the  cost  of  making,  together  with  a  sum,  the  interest  of 
which  would  defray  the  annual  repairs  and  expenses  of  the  road,  for 
the  distance  saved,  may  be  profitably  expended.  In  the  remaining  exam 
ples  of  this  section,  the  rate  of  interest  is  understood  to  be  6  per  cent. 


176  THE   ROAD.  [ART.  XL 

•j-1^  for  contractor's  profit,  the  earth  costing  10  cents  per 
c.  yd.  ?  Ans.  8324.79. 

10.  The  average  power  of  draft  of  a  horse,  moving  3  miles 
per  hour  for  10  hours  a  day,  being  100  lb.,  what  will  be  the 
annual  cost  of  transportation  over  a  road  30  miles  long,  on 
which  the  average  friction  is  ^  of  the  weight,  estimating 
the  ai    <unt  transported  at  50000  tons,  and  the  value  of  a 
days'  I  .oor  of  a  horse  at  75  cents  ?  Ans.  $42000. 

11.  If  the  road  in  the  preceding  example  should  be  im 
proved  by  macadamizing,  or  otherwise,  so  that  the  friction 
should  be  reduced  to  ^  of  the  weight,  what  would  be  the 
annual  cost  of  the  transportation,  and  how  much  might  be 
profitably  expended  in  making  the  improvement  ? 

2d  Ans.  $420000. 

12.  The  annual  cost  of  transportation  over  a  road  15  miles 
long,  being  estimated  at  $35000  per  mile,  what  amount  of 
saving  can  be  effected  by  expending  $20000  to  shorten  the 
road  2  miles,  and  $1000000  to  reduce  the  friction  to  J  its 
present  amount,  the  annual  cost  of  repairs  being  the  same 
in  both  cases?  Ans.  $3938333 £. 

13.  If  a  hill  by  friction  and  gravity,  causes  5000  days' 
work  of  a  horse,  at  75  cents  per  day,  which  can  be  avoided 
by  a  road  along  the  base  of  the  hill,  that  will  require  only 
2300  days'  work,  and  if  the  new  road  will  require  an  extra 
annual  outlay  of  $375  for  repairs,  how  much  can  be  saved 
by  expending  $10000  in  making  the  improvement  ? 

Ans.  $17500. 

14.  It  is  calculated  that,  in  locomotives,  the  evaporation  of 
1  cubic  foot  of  water  per  hour,  produces  a  mechanical  force 
of  about  two  horse  power,  and  that  a  horse  on  a  railway  can 
pull  10  tons  with  ease.1     According  to  this  estimate,  what 
load  can  be  drawn  by  a  locomotive  which  evaporates  175  c. 
ft  per  hour  ? 

*  Chambers. 


§50.] 


MISCELLANEOUS   EXAMPLES. 


177 


15.  If  an  engine  has  sufficient  force  to  draw  92T.  19cwt. 
Iqr.  over  level  ground,  what  additional  power  must  be  exerted 
on  an  ascending  grade  of  37ft.  per  mile?a 

Ans.  13cwt. 


16.  Determine  the  amount  of  excavation  and  embankment 
in  the  following  example,  by  taking  the  average  of  the  end 
areas  of  each  section  as  the  true  area  of  the  section/  and 
find  the  cost  of  the  whole  at  lOcts.  per  cubic  yard. 


Station. 

Distance. 

End  Areas. 

Excavation, 
Cubic  feet. 

Embankment. 
Cubic  feet. 

1 

2 
3 
4 

561  ft. 

858    < 
825    ' 

0 

1386  sq.  ft.  excav. 
1600  ' 
0  '             " 

388773 

'  b 

0 

o 

5 
6 

7 

820   < 
825    < 
330   « 

1672  '            emb. 

528  < 
0  '             " 

0 
0 
0 

685520 

2329767 

1680140 

Cost,  $14851.51. 


a  On  all  inclined  planes,  the  power  is  to  the  weight,  as  the  length  of 
the  plane  is  to  the  height. 

b  This  method,  which  is  the  one  usually  employed,  gives  a  result 
greater  than  the  true  contents.  Sometimes  the  calculation  is  performed 
by  deducing  the  middle  area  of  each  section  from  the  arithmetical  mean 
of  the  heights  at  the  two  extremities,  but  the  result  thus  obtained  is 
too  small.  The  true  contents  may  be  found  by  the 

PRISMOIDAL  FORMULA. 

Find  the  area  of  each  end  of  the  mass,  and  also  the  middle  area, 
corresponding  to  the  arithmetical  mean  of  the  heights  of  the  two  ends. 
Add  together  the  area  of  each  end,  and  four  times  the  middle  area. 
Multiply  the  sum  by  the  length,  and  divide  the  product  by  six.  The 
quotient  will  be  the  true  cubic  contents  required. 

The  reduction  of  the  contents  to  cubic  yards  would  be  greatly 
facilitated  if  the  distances  of  die  stations  were  always  some  multiple 
of  54  feet.— Gillespie. 

12 


178  THE   ENGINEER.  [ART.  XII. 

XII.    THE  ENGINEER. 
51.    THE  STEAM  ENGINE. 

A  POUND  of  steam  at  212°  will  raise  the  temperature  of 
a  pound  of  water  970°.a  But  as  some  of  the  heat  is  wasted, 
the  increased  temperature  may  be  considered,  in  practice,  as 
equivalent  to  900°.  From  this  fact  the  following  rule  is 
derived : — 

u  To  find  the  quantify  of  steam  required  to  raise  a  given 
quantity  of  water  to  any  required  temperature. — Multiply 
the  number  of  gallons  to  be  warmed  by  the  number  of 
degrees  between  the  temperature  of  the  cold  water  and  that 
to  which  it  is  to  be  raised,  for  a  dividend ;  and  to  the  excess 
of  the  temperature  of  the  steam  above  212°  add  900  for  a 
divisor.  The  quotient  will  be  the  number  of  gallons  formed 
into  steam,  required."5 

A  "  horse  power"  was  estimated  by  Boulton  and  Watt  as 
sufficient  to  raise  32000  Ib.  avoirdupois  1  foot  high  in  1 
minute,  but  in  estimating  the  force  of  their  engines,  they 
used  44000  as  a  divisor  instead  of  32000.  Desaguliers's 
estimate  was  27500 ;  Smeaton's,  22916 ;  some  of  the  modern 
English  engines  are  computed  at  66000,c  but  the  number 
commonly  used  is  33000,d  or  44000  when  an  allowance  of 
i  is  made  for  friction. 

To  calculate  the  power  of  an  engine.  Form  the  continued 
product  of  the  number  of  square  inches  in  the  area  of  the 
cylinder,  the  number  of  pounds  which  represents  the  effective 
pressure6  per  square  inch,  and  the  number  of  feet  through 
which  the  piston  moves  per  minute,  and  divide  by  the  number 
of  pounds  a  horse  can  raise  1  fooFper  minute. 

a  Daniell.  b  Pilkington.  c  Sci.  American. 

3  It  is  customary  to  consider  the  friction  of  the  machinery  as  equiva 
lent  to  I  of  the  effect  produced. 

•  The  effective  pressure  is  the  force  remaining  after  making  allow 
ance  for  the  waste  and  friction  of  the  steam. 


§51.]  THE   STEAM  ENGINE.  179 

The  following  rule  is  simpler,  and  in  most  cases  it  will  be 
found  sufficiently  accurate.  When  the  usual  estimate  of  a 
horse  power  is  employed,  and  the  effective  force  is  8.4  Ib.  per 
square  inch,  and  the  distance  traversed  by  the  piston  200ft. 
per  minute,  the  same  result  will  be  obtained  by  either  rule. 

Square  the  number  of  inches  in  the  diameter  of  the 
cylinder,  and  multiply  by  .04.  The  product  will  give  the 
number  of  horse  power. 

EXAMPLES. 

1.  What  quantity  of  water  converted  into  steam  at  220° 
will  raise  100  gallons  of  water  at  50°  to  the  boiling  point? 

Ans.  17|f4  gallons. 

2.  What  is  the  power  of  a  steam  engine  with  a  cylinder 
37  inches  in  diameter,  making  the  usual  estimate  of  the 
effective  force  of  the  steam  and  the  stroke  of  the  piston  ? 

Ans.  54.76  horse  power. 

3.  Find  by  each  rule,  the  power  of  a  steam  engine  that 
makes  10  strokes  per  minute,  each  stroke  being  8ft.,  allowing 
for  friction,  £  of  the  force,  which  is  15  Ib.  per  sq.  in.,  the 
diameter  of  the  cylinder  being  40  inches  ? a 

Ans.  by  Rule  1,  60.928  horse  power. 
"      "     2,  64  "         " 

4.  Two  steam  engines  constructed  for  the  island  of  Ceylon, 
working  10  hours  per  day  for  300  days  in  the  year,  will 
convert  576000  Ib.   of  paddyb  into  rice  worth  £116035, 
while  by  the  common  method,  the  same  quantity  of  paddy 
converted  into  rice,  would  yield  only  £64799.°     Allowing 
£20000  per  annum,  for  interest,  repairs,  and   the  extra 
expense  of  working  the  Machinery,  what  is  the  average  daily 
amount  saved  by  each  engine  ?          Ans.  104?.  2s.  4.8c?. 

a  As  the  piston  must  move  forward  and  back  at  each  stroke,  the  dis 
tance  passed  through  per  minute  is  160ft. 
b  Rice  in  the  husks.  c  Partington. 


180 


THE   ENGINEER. 


[ART.  XII. 


5.  An  engine  at  the  Wheal  Hope  mine,  in  Cornwall,  works 
3  pumps,  the  length   of  stroke  of  each  being  8ft. ;  their 
pistons  support  and  lift  at  each  stroke,  columns  of  water, 
whose  joint  weights  are  27766  lb.,  and  in  the  month  of 
December,  1826,  they  made  261890  strokes.4     Required 
the  velocity  per  minute,  and  the  total  dynamical  effect.6 

Ans.  Velocity  46.93ft. ;  effect  1303058. 38  lb. 

6.  The  working  effect  of  1  bushel  of  coals,  or  the  number 
of  pounds  which  could  be  raised  1ft.  high  by  1  bushel  is 
called  the  Duty  of  the  engine.     Required  the  duty  of  the 
engine  in  the  preceding  example,  the  amount  of  coal  con 
sumed  being  1242  bushels.  Ans.  46838246. 

7-10.  A  cubic  foot  of  water  makes  1689  c.  ft.  of  steam, 
at  the  temperature  of  212°. c  Estimating  the  pressure  of  the 
atmosphere  at  2120  lb.a  on  a  square  foot,  what  will  be  the 
dynamical  effect  of  1  lb.  of  each  of  the  following  kinds  of 
fuel?d 


Fuel. 

Spec,  gravity. 

Weight  to  evaporate 
1  c.  ft.  of  water. 

Dynamical  effect 
of  1  lb.  of  fuel. 

Anthracite  coal 
Va.  bituminous  coal 
Pa. 
Dry  pine  wood 

1500 
1364 
1262 
336 

6.534  lb. 
7.82    « 
9.37    " 
15.4      « 

548007  lb.  1ft. 
lb.   " 
lb.  " 
232512  lb.  « 

THE  WATER-WHEEL. 

The  pressure  of  water  on  any  surface  is  equal  to  the 
weight  of  a  column  of  water  with  the  same  base  as  the 
surface  pressed,  and  of  a  height  equivalent  to  the  depth  of 
the  centre  of  gravity.  If  the  surface  is  of  any  regular 
shape,  the  centre  of  gravity  corresponds  with  the  centre  of 
the  surface. 

The  actual  velocity  of  water  flowing  from  an  orifice,  or 

*  Moseley. 

b  The  dynamical  effect  is  found  by  multiplying  the  velocity  by  the 
weight. 
c  Daniell.  d  W.  R.  Johnson. 


§52.]  THE    WATER-WHEEL.  181 

falling,  will  generally  be  .6  or  .7  of  the  theoretic  velocity.* 
To  obtain  the  theoretic  velocity  per  second,  find  the  number 
of  feet  in  the  perpendicular  fall  of  the  water,  (or  in  the 
depth  below  the  surface  to  the  middle  of  the  orifice,)  extract 
the  square  root,  and  multiply  by  8.018. 

The  effective  force  of  a  water-wheel,  may  be  estimated  at 
f  of  the  power  applied  to  it,  if  the  wheel  is  overshot,  and 
at  i  of  the  power  if  the  wheel  is  undershot.5  To  determine 
the  power  of  a  stream  of  water,  measure  the  breadth  and 
depth  of  the  stream,  the  height  of  fall,  and  the  velocity  per 
minute,  all  in  feet;  form  the  continued  product  of  these 
four  numbers,  and  divide  by  792. c  For  undershot  wheels 
take  £  this  result. 

To  calculate  the  power  of  machinery  or  wheelwork, 
multiply  together  the  effective  power,  and  the  lengths  of  all 
the  driving  levers,  (or  the  radii,  circumferences,  cogs  or 
rounds  of  the  driving  wheels,)  and  divide  by  the  continued 
product  of  the  lengths  of  all  the  leading  levers,  (or  the  radii, 
&c.,  of  the  leading  wheels.)  For  the  velocity,  multiply  the 
velocity  of  the  power  by  the  lengths,  radii,  circumferences, 
cogs  or  rounds  of  the  leading  levers  or  wheels,  and  divide 
by  the  product  of  the  like  dimensions  of  all  the  driving 
levers  or  wheels. 

The  maximum  effect  will  be  produced  in  machinery  of 
any  kind,  when  the  load,  or  resistance,  is  |  of  the  power,  and 
when  the  velocity  of  the  machinery  at  the  point  of  action, 
is  |  of  the  greatest  velocity  of  the  power.d 

EXAMPLES. 

1.  What  is  the  amount  of  pressure  on  a  dam  75ft.  by 
12ft.,  the  depth  of  water  being  8ft.  ?e 

Am.  225000  Ib. 

a  Nicholson.  b  Pjlkington. 

c  ?  *  M:^:  v  Rf>  ,  _  D.B.H.V. 

3  33000      X  793 

d  Brande.  e  The  depth  of  the  centre  of  gravity  is  4ft. 


182  THE   ENGINEER.  [ART.  XII, 

2.  What   is   the   theoretical  velocity  of  water  flowing 
through  an  orifice,  the  centre  of  which  is  6ft.  Sin.  beloiv 
the  surface  ?  Ans.  20.045ft.  per  second. 

3.  Find  the  bottom  pressure,  and  the  entire  pressure  upon 
the  bottom  and  sides  of  a  cube,  each  side  of  which  measures 
8  feet.  Ans.  Bottom  pressure,  32000  Ib. 

Entire         "         96000  Ib. 

4.  If  the  area  of  an  orifice  is  2£  sq.  ft.,  and  the  velocity 
of  the  water  flowing  through  it  is  15ft.  per  second,  what  will 
be  the  weight  of  the  water  discharged  in  1  minute? 

Ans.  131250  Ib. 

5.  A  stream  12in.  deep,  and  22in.  broad,  moves  with  a 
velocity  of  88ft.  in  15".     Required  its  effective  force,  with 
a  fall  of  50ft.  Ans.  40|Q  horse  power. 

6.  How  many  cuts  are  made  per  minute  by  the  beater  of 
a  paper  mill,  which  has  60  teeth,  each  of  which  passes  by 
24  cutters  at  every  revolution,  when  there  are  150  revolutions 
per  minute  ?  Ans.  216000  cuts.a 

7.  What  force  will  be  exerted  at  the  distance  of  2  feet 
from  the  centre   of  a  millstone,  by  a  water-wheel  with  a 
power  of   1000  Ib.,   the  diameters   of  the  driving  wheels 
being  8ft.,  2ft.,  and  1ft.,  and  the  diameters  of  the  leading 
wheels  being  4ft.  and  3ft.  ?  Ans.  666f  Ib. 

8.  If  the  power  in  the  preceding  example  moved  with  a 
velocity  of  12ft.  per  second,  what  would  be  the  velocity  of 
the  millstone  ?  Ans.  18ft. 

9.  What  should  be  the  area  of  the  section  of  a  canal,  to 
deliver  90000  c.  ft.  per  hour,  the  water  moving  with  a 
velocity  of  4ft.  per  second  ?  Ans.  6i  sq.  ft. 

10.  Using  10  c.  ft.  per  second,  what  time  would  be  neces 
sary  to  exhaust  a  pond,  the  area  of  which  is  10  acres,  and 

a  This  rapid  motion  makes  a  coarse  musical  note,  that  can  be  heard 
at  a  great  distance  from  the  mill. —  Ure. 


§53.]  PUMPS.  183 

the  average  depth  3  feet,  the  pond  being  supplied  by  a 
stream  that  furnishes  150  c.  ft.  per  minute  ? 

Ans.  48h.  24min. 

53.    PUMPS." 

The  power  employed  in  working  a  pump  is  estimated  by 
multiplying  the  number  of  pounds  discharged  per  minute, 
by  the  number  of  feet  that  the  water  is  raised  above  the 
reservoir. 

The  weight  of  water  in  a  yard  of  pipe  may  be  found  very 
nearly  by  squaring  the  number  of  inches  in  the  diameter  of 
the  pipe,  and  increasing  the  square  by  -fa  of  itself.b  The 
result  will  be  the  weight  in  pounds  avoirdupois. 

The  number  of  ale  gallons  in  a  yard  of  pipe  may  be  found 
very  nearly  by  squaring  the  number  of  inches  in  the  diameter, 
and  dividing  by  10. 

In  estimating  the  power  necessary  to  overcome  resistance 
in  pumps,  -J-  should  be  added  for  the  friction  of  the  water. 

The  diameter  of  the  pipes  should  be  at  least  as  great  as 
the  diameter  of  the  pump.  If  it  is  greater,  the  friction  will 
be  diminished. 

EXAMPLES. 

1.  At  the  height  of  225  ft.  above  the  level  of  a  reservoir, 
250  ale  gallons  are  discharged  per  minute.     Required  the 
power  of  the  engine,  the  weight  of  one  ale  gallon  of  water 
being  10Hb.c  Ans.  13.1  horse  power. 

2.  Find  the  weight  of  water,  and  the  number  of  ale  gal 
lons,  in  a  pipe  15  rods  long,  and  3  inches  in  diameter. 

Ans.  759|  Ib. ;  74*  gallons. 

3.  On  the  top  of  a  hill  75  feet  high,  is  a  reservoir  40  feet 

a  Nicholson,  Pilkington,  Ferguson. 
0.7854X3X62.5        45 
144  44 

c  In  all  the  examples  of  this  section,  an  allowance  of  i  is  made  for 
the  friction  of  the  machinery.  See  Sect.  51. 


184  THE   LABORATORY.  [ART.  XIII. 

square,  and  12ft.  deep.     What  power  is  necessary  to  fill 
the  cistern  in  45  minutes  ? 

40x40x12x62.5x75     6     K, 
AnS  ---  45X44000        -xr»4/T  horse  power. 

4.  What  power  is  necessary  to  fill  a  cistern  30ft.  long, 
22ft.  wide,  and  10ft.  deep,  in  25  minutes,  the  water  being 
raised  100ft.  ?  Am.  45  horse  power. 

5.  What  should  be  the  diameter  of  the  pump  in  each  of 
the  two  preceding  examples,  if  there  are  40  strokes  per 
minute,  the  length  of  the  effective*  stroke  being  2ft.  ? 

Ans.  Ex.  3  :   40x40x12  =  426|  c.  ft.  per  minute  ; 
45 

426f  -^  (2  X  40)  =  5£  sq.  ft.  area  of  pump  ; 


-f-  .7854  =  2.6ft.  diameter  of  the  pump. 
Ex.  4  :    Diameter  2.05ft. 

6.  A  town  of  25000  inhabitants,  is  to  be  supplied  with 
water  from  a  river  200  feet  below  the  proposed  reservoir. 
Estimating  the  average  daily  consumption  at  9  ale  gallons 
for  each  individual,  what  must  be  the  power  of  an  engine 
working  10  hours  per  day,  and  what  will  be  the  size  of  the 
pump,  making  30  strokes  per  minute,  the  effective  stroke 
being  3ft.  ?  Ans.  Engine  17.47  horse  power. 

Area  of  pump  98  sq.  in.  nearly. 

Diam.  of  pump  11.17in. 


XIII.    THE  LABORATORY. 

CHEMICAL  COMBINATIONS.*" 


IN  chemical  compounds,  the  following  curious  facts  have 
been  observed. 

*  An  allowance  is  made  from  the  stroke  of  the  piston  rod  for  the 
escape  of  water  through  the  valves.  Pilkington  states  this  allowance 
at  3  inches. 

b  Draper. 


§54.]  CHEMICAL   COMBINATIONS.  185 

1.  The  constitution  of  a  compound  body  is  always  the 
same.     Thus  it  has  been  found  that  9  grains  of  water  con 
tain  8  grains  of  oxygen  and  1  grain  of  hydrogen;    and 
however  often  the  analysis  is  repeated,  this  proportion  is 
found  to  be  invariable. 

2.  The  proportions  in  which  bodies  are  disposed  to  unite 
with  each  other,  can  always  be  represented  by  certain  num 
bers.     Thus  water  is  composed  of  one  atom  of  oxygen  and 
one  atom  of  hydrogen,  and  as  the  oxygen  atom  is  8  times 
as  heavy  as  that  of  hydrogen,  it  follows  that  in  9  parts  by 
weight,  of  water,  there  are  8  parts  of  oxygen  and  1  of 
hydrogen. 

3.  If  two  substances  unite  with  each  other  in  more  pro 
portions  than  one,  those  proportions  bear  a  simple  arithmeti 
cal  relation  to  each  other;  thus  14  grains  of  nitrogen  will 
successively  unite  with  8,  16,  24,  32,  40  grains  of  oxygen, 
forming  five  different  compounds,  which  contain  respectively 
1  atom,  2  atoms,  3  atoms,  4  atoms,  5  atoms  of  oxygen,  and 
1  atom  of  nitrogen. 

There  are  three  ways  in  which  the  composition  of  a  sub 
stance  may  be  expressed :  1,  by  atom  ;  2,  by  weight ;  3,  by 
volume.  Thus  water  is  composed,  by  atom,  of  oxygen  1, 
and  hydrogen  1;  by  weight,  of  hydrogen  1,  and  oxygen  8; 
and  by  volume,  of  hydrogen  2,  and  oxygen  1. 

Elementary  bodies  are  represented  in  chemistry  by  letters 
or  symbols.  A  list  of  the  elements  and  symbols  is  given 
in  the  next  section. 

A  symbolic  letter  standing  alone,  represents  one  atom  of 
the  element.  Thus  C  denotes  one  atom  of  carbon ;  0,  one 
atom  of  oxygen. 

To  denote  more  than  one  atom,  we  may  either  repeat  the 
symbol,  or  a  figure  may  be  placed  either  before  or  after  the 
symbol :  thus  000,  30,  or  03  would  each  represent  3 


186  THE   LABORATORY.  [ART.  XIII. 

atoms  of  oxygen.  The  latter  method  is  usually  adopted. 
Nitric  acid,  which  is  composed  of  1  atom  of  nitrogen  and 
5  of  oxygen,  is  denoted  by  N05. 

To  denote  a  compound  formed  of  several  compounds,  we 
employ  one  or  more  commas,  thus :  S03,HO,  which  is  the 
formula  of  strong  oil  of  vitriol. 

The  terms,  combining  proportion  and  chemical  equivalent, 
have  the  same  meaning  as  atomic  weight. 

55.    TABLE  or  CHEMICAL  EQUIVALENTS.* 


Names  and  Symbols  of 
Elements. 

Hydro- 
gen=  1. 

Names  and  Symbols  of 
Elements. 

Hydro- 
gen=l. 

Aluminum 

Al 

13.72 

Manganese 

Mn 

27.72 

Antimony 

Sbb 

129.24 

Mercury     . 

Hgh 

101.43 

Arsenic   . 

As 

75.34 

Molybdenum 

Mo 

47.96 

Barium    . 

Ba 

68.66 

Nickel    . 

Nk 

29.62 

Bismuth  . 

Bi 

71.07 

Nitrogen 

N 

14.19 

Boron 

B 

10.91 

Osmium 

Os 

99.72 

Bromine  . 

Br 

78.39 

Oxygen  . 

0 

8.01 

Cadmium 

Cd 

55-83 

Palladium 

Pd 

53.36 

Calcium  . 

Ca 

20.52 

Phosphorus 

P 

31.44 

Carbon    . 

C 

6.04 

Platinum 

Pt 

98.84 

Cerium    . 

Ce 

46.05 

Potassium 

KI 

39.26 

Chlorine  . 

Cl 

35.47 

Rhodium 

R 

52.2 

Chromium 

Cr 

28.19 

Selenium 

Se 

39.63 

Cobalt     . 

Co 

29.57 

Silicon    . 

Si 

22.22 

Columbium 

Tac 

184.9 

Silver     . 

Agk 

108.31 

Copper    . 

Cud 

31.71 

Sodium  . 

Na1 

23.31 

Didymium 

D 

9 

Strontium 

Sr 

43.85 

Erbium    . 

E 

9 

Sulphur 

S 

16.12 

Fluorine  . 

Fl 

18.74 

Tellurium 

Te 

64.25 

Glucinum 

G 

26.54 

Terbium 

Tr 

? 

Gold   .     . 

Aue 

199.2 

Thorium 

Th 

59.83 

Hydrogen 
Iodine 

H 
I 

1. 

126.57 

Tin     .     . 
Titanium 

Snm 
Ti 

58.92  ! 
24.33 

Iridium    . 

Ir 

98.84 

Tungsten 

Wn 

94.8 

Iron    . 

Fef 

27.18 

Vanadium 

V 

68.66 

Lantanum 

La 

? 

Uranium 

U 

217.2 

Lead   .     . 

Pb* 

103.73 

Yttrium  . 

Y 

32.25 

Lithium   . 

Li 

6.44 

Zinc   .     . 

Zn 

32.31 

Magnesium 

Ma 

12.89 

Zirconium 

Z 

33.67 

a  Compiled  from  Parnell  and  Draper.        b  Stibium.        c  Tantalum. 
(1  Cuprum.  e  Aurum.  f  Ferrum.  &  Plumbum. 

h  Hydrargyrum.       '  Kalium.  k  Argentum.  '  Natrium. 

m  Stannum.  n  Tungsten  or  Wolfram. 


§56.]  MISCELLANEOUS   EXAMPLES.  187 

If  compounds  are  united  by  a  feeble  affinity,  the  sign  't 
is  sometimes  used.  Thus  the  composition  of  sulphuric  acid 
may  be  indicated  by  S03,  or  by  S02  +  0,  the  latter  formula 
showing  that  one  of  the  atoms  of  oxygen  is  held  by  a  feebler 
affinity  than  the  other  two. 

1-58.  Find  the  value  of  each  of  the  foregoing  equivalents, 
assuming  oxygen  =  100.  Ans.  Al  171.28. 

Sb  1613.48,  &c.  &c. 

56.     EXAMPLES  FOR  THE  PUPIL. 

1-33.  Find  the  atomic  weights*  of  each  of  the  following 
Acids  :b  Acetic,  C4H303;  Arsenic,  As05;  Arsenious,  As03j 
Benzoic,  C14H503;  Boracic,  B03;  Bromic,  Br05;  Carbonic, 
C02;  Chloric,  C105;  Chromic,  Cr03;  Citric,  C4H204;  For 
mic,  C2H03;  Gallic,  C7H03;  Hydriodic,  HI;  Hydrobromic, 
HBr;  Hydrochloric,  HC1;  Hydrocyanic,  H  +  C2N;  Hydro 
fluoric,  HF1 ;  Hydrosulphuric,  HS ;  Hypermanganic,  Mn207; 
Hyposulphurous,  S202;  Hyposulphuric,  S205;  lodic,  I05; 
Lactic,  C6H505;  Malic,  C8H408;  Manganic,  Mn03;  Nitric, 
X05;  Oxalic,  C203;  Phosphoric,  P05;  Silicic,  Si03;  Sul 
phuric,  S03;  Sulphurous,  S02;  Tannic,  C18H509;  Tartaric, 
C8H4010. 

34-58.  Find  the  atomic  weights  of  each  of  the  following 
Bases  :  Alumina,  A1203 ;  Ammonia,  NH3 ;  Oxide  of  Anti 
mony,  Sb03 ;  Barytes,  BaO ;  Oxide  of  Chromium,  Cr203 ; 
Oxide  of  Cobalt,  CoO;  Protoxide  of  Copper,  CuO;  Sub- 

a  The  equivalent  of  a  compound  body  is  always  equal  to  the  sum  of 
the  equivalents  of  its  constituents. — Parnell. 

b  "  Compound  bodies  may,  for  the  most  part,  be  divided  into  three 
groups  ;  acids,  bases,  and  salts.  By  an  acid,  we  mean  a  body  having 
a  sour  taste,  reddening  vegetable  blue  colors,  and  neutralizing  alkalies  ; 
by  a  base,  a  body  which  restores  to  blue  the  color  reddened  by  an  acid, 
and  possessing  the  quality  of  neutralizing  the  properties  of  an  acid  ;  by 
n  salt,  the  body  arising  from  the  union  of  an  acid  and  a  base.  These 
definitions,  however,  are  to  be  received  with  considerable  limitation." 
— Draper. 


188  THE   LABORATORY.  [ART.  XIII. 

oxide  of  Copper,  Cu20 ;  Peroxide  of  Iron,  Fe203  ;  Protoxide 
of  Iron,  FeO ;  Protoxide  of  Lead,  PbO ;  Lime,  CaO ;  Mag 
nesia,  MaO;  Protoxide  of  Manganese,  MnO;  Oxide  of 
Mercury,  HgO;  Suboxide  of  Mercury,  Hg20;  Oxide  of 
Nickel,  NiO ;  Oxide  of  Platinum,  PtO ;  Potash,  KO ;  Ox 
ide  of  Silver,  AgO ;  Soda,  NaO ;  Strontian,  SrO  ;  Protoxide 
of  Tin,  SnO;  Peroxide  of  Tin,  Sn02;  Oxide  of  Zinc,  ZnO. 

59.  Hydrochloric  acid  consists  of  equal  volumes  of  chlo 
rine  and  hydrogen,  united  without  condensation.     Required 
its  specific  gravity,  the   specific  gravity  of  hydrogen  being 
.069,  and  that  of  chlorine  2.47.a  Ans.   1.2695. 

60.  If  one  volume  of  carbonic  acid  gas  contains  1  volume 
of  oxygen  and  1  volume  of  carbon  vapor,  what  is  the  spe 
cific  gravity  of  carbon  vapor,  the  specific  gravity  of  carbonic- 
acid  being  1.5238,  and  that  of  oxygen  being  1.1025? 

Ans.  .4213. 

61.  The  vapor  of  alcohol  is  composed  of  8  volumes  of 
carbon,  12  volumes  of  hydrogen,  and  2  volumes  of  oxygen, 
the  whole  being  condensed  into  4  volumes  of  vapor,     lie- 
quired  its  specific  gravity ;  the  specific  gravity  of  carbon 
being  .4213,  that  of  hydrogen  .069,  and  that  of  oxygen 
1.1025.  Ans.  1.60085. 

62.  According  to  the  experiments  of  Despretz,  loz.  of 
carbon  evolves,  during  its  combustion,  as  much  heat  as  would 
raise  the  temperature  of  loz.  of  water  14067°.     How  many 
pounds  of  water  could  be  raised  from  the  freezing  point  to 
the  mean  temperature  of  the  human  body,  (98.3°,)  by  the 
13.9oz.  of  carbon,  which  are  daily  converted  into  carbonic 
acid  in  the  body  of  an  adult  ?b  Ans.  184.3  Ib. 

*  In  determining  the  specific  gravity  of  gases,  air  is  generally  assumed 
as  the  standard,  =  1. 
b  Liebig. 


§57.]  GENERAL   REMARKS.  189 

XIV.    GENERAL  ANALYSIS. 
57.    REMARKS  ON  THE  SOLUTION  OF  QUESTIONS. 

IN  most  treatises  on  arithmetic,  after  the  fundamental 
rules  have  been  taught,  the  various  applications  of  those 
rules  are  arranged  under  different  heads,  such  as  Interest, 
Discount,  Practice,  Proportion,  Profit  and  Loss,  Fellowship, 
Bankruptcy,  &c.  This  division  of  the  subject  is  convenient 
for  beginners,  but  the  expert  arithmetician  must  be  entirely 
independent  of  formal  rules  ;  he  must  be  able,  by  analyzing 
any  question  that  is  proposed  to  him,  to  determine  what  ope 
rations  are  necessary  for  its  solution.  Accountants,  men  of 
business,  and  nearly  all  who  are  required  to  make  frequent 
calculations,  perform  most  of  their  work  by  analytical  proc 
esses,  and  not  by  the  rules  that  they  learned  at  school. 

Mistakes  are  rarely  made  in  determining  the  proper  mode 
to  be  pursued  when  numbers  are  to  be  merely  added  or  sub 
tracted  ;  but  when  multiplication  or  division  is  required, 
care  is  often  necessary  to  avoid  multiplying  when  we  ought 
to  divide,  or  dividing  when  we  ought  to  multiply.  Practice 
and  careful  attention  to  the  conditions  of  the  question  will 
generally  remove  all  difficulty.  Much  unnecessary  labor 
may  often  be  avoided,  by  stating  the  operations  as  we  pro 
ceed,  and  not  performing  any  of  the  multiplications  or 
divisions  until  the  statement  is  complete.  Thus,  if  it  were 
required  to  multiply  27  by  19,  to  divide  the  product  by  2 
times  171,  and  to  multiply  the  quotient  by  f  of  J  of  |  of 
594,  the  readiest  method  of  arriving  at  the  result,  would  be 

97  v  1  Q        9  v  7  v  4  v  ^Qd 
to  express  the  operations  thus,     '    ,    ,  X 

' 


,    , 
x171      3x8x9x1      ' 

then  by  cancelling,  we  readily  determine  the  answer,  which 
is  7x33  or  231. 

One  of  the  two  following  methods,  will  furnish  the  answer 
to  nearly  all  questions  that  admit  of  an  analytical  solution. 


190  GENERAL  ANALYSIS.        [ART.  XIV. 

1st.  In  the  majority  of  cases,  we  should  endeavor  to  find 
from  the  terms  that  are  given,  the  value  which  would  cor 
respond  to  ONE  of  each  of  the  terms  concerning  which  an 
answer  is  required.  Having  found  the  answer  for  ONE,  we 
can  easily  determine  it  for  the  REQUIRED  NUMBER  in  each  of 
the  terms  of  demand. 

2d.  "We  are  sometimes  required  to  reason  from  a  result  to 
its  origin.  In  such  cases  it  is  generally  best  to  reverse  all 
the  operations  by  which  the  result  was  obtained.  Examples 
illustrating  both  of  these  methods,  will  be  found  in  the 
two  following  sections. 

Whenever  fractions  or  decimals  are  involved  in  the  condi 
tions  of  a  question,  the  same  operations  must  be  performed 
on  them,  that  would  be  required  if  their  place  were  occupied 
by  whole  numbers.  Therefore,  if  we  are  ever  at  a  loss 
whether  to  multiply  or  divide  by  a  fraction,  all  difficulty  may 
be  removed  by  substituting  a  small  integral  number,  and 
considering  what  would  then  be  required. 

«5§.    EXAMPLES  ILLUSTRATING  THE  FIRST  METHOD. 

The  pupil  should  be  required  to  repeat  the  analysis  of 
all  the  following  examples,  and  encouraged  to  give  a  solution 
of  his  own,  whenever  a  different  one  occurs  to  him. 

1.  What  is  the  interest  of  $635.50  for  3y.  8mo.  24dy.  at 
6  per  cent.  ? 

The  interest  of  ONE  dollar  for  ONE  year,  is  $.06.     For  3y.  8mo. 

24dy.  it  would  be  3J \  times  as  much,  or  $56  X  .06     The  interest 

15 

of  $635.50  will  be  635.50  times  as  much  as  the  interest  for  ONE 
dollar,  or  ?635.50X  66  X  .06»=  $  ? 

15 

a  The  object  of  these  examples  is  merely  to  show  how  the  general 
principles  of  analysis  can  be  applied  to  all  classes  of  questions.  All 
the  short  processes  and  contractions  that  the  pupil  may  have  learned, 
and  can  retain  in  his  memory,  he  may  be  allowed  to  employ. 


§58.]  MISCELLANEOUS   EXAMPLES.  191 

2.  How  much  per  cent,  is  gained  by  selling  at  9  Gets,  a 
pound,  tea  that  cost  75  cents  ? 

A  gain  of  ONE  per  cent,  on  a  pound,  would  be  $.0075.  The 
actual  gain  was  15  cents,  which  is  (.15  .4-  .0075)  times  ONE  per 
cent.,  or  per  cent.  ? 

3.  If  $400  at  simple  interest  amounts  to  $440.50  in  2y. 
3mo.,  what  is  the  rate  per  cent.  ? 

The  interest  is  $40.50.  The  interest  of  ONE  dollar  for  ONE  year 
at  ONE  per  cent.,  would  be  $.01.  The  interest  of  $400  for  2Jy. 

at  ONE  per  cent.,  =    _  X  -  X  —      ^ne  NUMBER  of  per  cent.  = 
141 

40.50        141 


Give  an  analytical  solution  of  the  above  question,  by  first 
finding  the  interest  for  one  year. 

4.  In  what  time  will  $900  amount  to  $1044  at  6  per 
cent.,  simple  interest  ? 

We  wish  to  find  in  how  many  years  $900  will  yield  $144  interest, 
at  6  per  cent.  In  ONE  year  it  would  yield  $54.  The  NUMBER  of 
years  is  therefore  144  -f-  54  =  yr.  ? 

5.  If  a  man's  property  yields  5£  per  cent,  simple  interest, 
and  his  annual  income  is  $1093.12  J,  what  is  he  worth  ? 

We  wish  to  find  how  many  dollars  he  is  worth.  If  he  was  worth 
ONE  dollar,  his  income  would  be  $.05£.  He  is  therefore  worth  as 
many  times  ONE  dollar,  as  are  equivalent  to  1093.12J-J-  .05J, 
or$  ? 

6.  "When  money  yields  but  5  per  cent,  interest,  what  is 
the  present  worth  of  $5316.84  due  in  lyr.  7mo.  6dy.  ? 

We  wish  to  find  how  many  dollars  will  amount  to  $5316.84  in 
1|  yr.  at  5  per  cent.  ONE  dollar  would  amount  to  $1.08.  The 
number  of  dollars  required,  is  therefore  as  many  times  ONE  dollar, 
as  are  equivalent  to  $5316.84  -j-  $1.08,  or  $  ? 

7.  What  must  be  the  face  of  a  note  at  90  days,  to  be  dis 
counted  at  Bank,  at  6  per  cent.,  to  yield  $500? 

We  wish  to  find  how  many  dollars  would  yield  $500,  and  we 


192  GENERAL  ANALYSIS.        [ART.  XIV. 

first  find  that  ONE  dollar  would  yield  $.9845.  The  number  of 
dollars  required  is  therefore  as  many  times  ONE  dollar,  as  are 
equivalent  to  500  -j- .  9845  =  $  ? 

8.  If  I  gain  25  per  cent,  on  the  original  cost,  by  selling 
merchandise  for  $1718.75,  how  much  did  it  cost  me? 

How  many  dollars  did  it  cost  ?  IF  it  had  cost  ONE  dollar,  to  gain 
25  per  cent,  it  must  have  been  sold  for  $1.25.  It  must  therefore 
have  cost  (1718.75-^-1.25)  times  ONE  dollar,  or  $  ? 

9.  When  exchange  on  England  is  at  a  premium  of  8  J  per 
cent.,  what  is  the  value  in  sterling  money,  of  $112  ? 

We  cannot  so  easily  find  the  value  of  $1  in  English  Money,  as 
the  value  of  £1  in  Federal  Money.1  If  ONE  £  =  $1.08  J  X  -4g°->  how 
many  £  =$112?  As  many  times  ONE  pound,  as  there  are  times 

1.08^  X4o in  112.     ^X2^7*®  =  £  ? 

10.  If  36  men,  in  127  J  days  of  13  J  hours,  dig  a  trench 
33f  yd.  long,  lOJft.  deep,  and  ISjjft.  wide,  how  many  men 
in  7f  days  of  12T8f  hours,  will  dig  a  similar  trench  82^Tyd. 
long,  7§ft.  deep,  and  10ft.  wide  ? 

First,  to  find  how  many  men  in  ONE  day,  of  ONE  hour,  would 
dig  a  trench  ONE  yd.  long,  ONE  ft.  deep,  and  ONE  ft.  wide,  we  must 
multiply  36  by  127 J  and  by  18  *,  and  divide  by  33|,  10  J,  and  16|, 

or  ¥  x  ^  x  2/  x  T-f  5  x  2T  x  ?V  Second> ia  7f  dars  u  win 

not  require  so  many  men  as  in  one  day;  working  12y8j-  hours  a 
day,  will  not  require  so  many  men  as  working  one  hour  a  day, 
&c.  ;  we  must  therefore  divide  our  first  result  by  7^  and  12  j8f, 
and  multiply  by  82 ,8r,  7§ ,  and  10,  which  will  give  3r6  X  -£-  X 
27  v  _4_  v  _2_  v  _5^  v  _7_  v  JUL  v  9-J-°  v  37  v  L°  men 

2Xl35*;2lX7gA51*140X     n     X     5     X     i 

11.  How  many  square  yards  in  a  room  28ft.  4in.  wide, 
and  37ft.  6in.  long? 

The  answer  is  to  be  in  yards,  and  therefore  the  dimensions  must 
be  reduced  to  yards.  The  width  is  9|yd.,  and  the  length  12,2yd. 
If  the  room  was  ONE  yard  long  and  ONE  yard  wide,  it  would  con- 

*  Some  attention  will  often  be  required,  to  determine  which  of  the 
quantities  should  be  taken  to  find  the  value  of  ONE. 


§58.]  MISCELLANEOUS   EXAMPLES.  193 

tain  one  square  yard.  But  the  width  is  9|yd.,  instead  of  lyd.  ; 
and  the  length  12  .?,yd.,  instead  of  lyd.  We  must  therefore  multiply 
1X895X~25=~  sq.  yd.? 

12.  If  a  man  receives  $30  for  building  8  rods  of  wall, 
and  he  can  purchase  3  barrels  of  flour  for  $14,  and  5cwt. 
of  sugar  for  G  barrels  of  flour,  and  126  Ib.  of  tea  for  lOcwt. 
of  sugar,  how  many  pounds  of  tea  can  he  purchase  by  build 
ing  24  rods  of  wall  ? 

We  cannot  find  directly  how  many  pounds  of  tea  he  can  pur 
chase  by  ONE  rod  of  wall,  but  for  ONE  rod  he  will  receive  $^>- 
For  ONE  dollar  he  can  buy  y3^  of  a  barrel  of  flour,  and  for  $3g° 
he  can  buy  3g°  x  -j\  °f  a  barrel.  For  ONE  barrel  of  flour  he  can 
buy  |cwt.  of  sugar,  and  for  3gO  x  T3^  bbl.,  he  can  buy  3go  x  _s_  x 
|cwt.  For  ONE  cwt.  of  sugar  he  can  buy  l~2-^  Ib.  of  tea,  and  for 
3g°  X  A  X  gcwt.  he  can  buy  3g°  X  &  X  |.X  W  lb-,  which  is 
therefore  equivalent  to  1  rod  of  wall,  and  24  rods  will  purchase 
¥  X  3g0  X  A  X  |  X  W  =  Ib.  of  tea? 


13.  A.'s  stock  in  a  partnership  is  $450,  B.'s  $350,  and 
C/s  $500.     How  must  a  loss  of  $169  be  divided  between 
them  ? 

If  $1300  loses  $169,  how  much  does  ONE  dollar  lose?  If  ONE 
dollar  loses  $  JQfo,  $450  will  lose  $1  f  &  X  -f/^  =  $  ,  $350  will 
lose  $^f£  X  TV\&=  $  ,  and  $500  will  lose  $^XTWa 

=  $         ? 

14.  Divide  650  into  four  parts,  which  shall  be  to  each 
other  in  the  proportion  of  *>  3?  |>  and  y7^. 

If  one  part  has  J  of  a  share,  another  £  of  a  share,  another  f 
of  a  share,  and  the  other  T7o  of  a  share,  then  the  whole  will  be 
2£  shares.  ONE  share  is  therefore  G50  -f-  2£.  Then  £  share  = 

i  X  ^  X  T63  =      >  i  share  =  }  X  *-{-£  X  T65  =       ,  |  share  = 
I  X  &6i*  X  -fs  =          ,  and  /3  share  =  T72  X  *%*  X  T63  =  ? 

15.  A  bankrupt  owes  $15600,  and  his  property  is  worth 
only  $10600.     How  much  can  he  pay  on  a  debt  of  $450  ? 

13 


194  GENERAL   ANALYSIS.  [ART.  XIV. 

On  ONE  dollar  he  can  pay  $|J888  =  ffSj  and  on  $450  he  can 


16.  A.  invests  a  certain  sum  for  a  certain  time,  B.  twice 
as  much  for  -J  of  the  time,  and  C.  three  times  as  much  for 
I  of  the  time.     How  should  they  share  the  gain,  which 
was  $559? 

Find  the  value  for  ONE  sum  for  ONE  time,  which  will  be  A.'s 
share.  2  sums  for  3  of  a  time  =  f  sum  for  ONE  time  =  B.'s  share. 
3  sums  for  |  of  a  time  =  1  -|  sums  for  ONE  time  =  C.'s  share.  Then 
the  whole  is  equivalent  to  1  •+-  f  +  lj  =  2  j  f  sums  for  ONE  time. 
ONE  sum  for  ONE  time  =  §559  -$-  2  \  |  =  $  ,  A.'s  share;  2  sums 
for  *  of  a  time,  or  |  of  a  sum  for  ONE  time  =  $|  X  -f-  X  If  == 
$  ,  B.'s  share;  3  sums  for  §  time,  or  1-  sums  for  ONE  time  = 
*Sx*fJlXiS=$  ,  C.'s  share. 

17.  A  goldsmith  mixed  3  Ib.  of  gold  22  carats  fine,  5  Ib. 
20  carats  fine,  8  Ib.  24  carats  fine,  and  4  Ib.  of  alloy.     What 
was  the  fineness  of  the  mixture  ? 

We  wish  to  find  how  many  carats  there  are  in  ONE  pound.  There 
are  20  Ib.  in  the  whole,  containing  carats,  and  therefore  in 
each  pound  there  are  -r-  20  =  carats  ? 

18.  Find  the  equated  time  for  the  payment  of  $500  due 
in  3  months,  and  $700  due  in  5  months. 

The  question  is,  in  how  many  months  should  $1200  be  paid,  and 
we  first  find  how  many  dollars  could  be  used  for  ONE  month,  to  be 
equivalent  to  the  two  debts.  If  $5000  could  be  used  ONE  month, 
how  many  months  could  $1200  be  used  ? 

19.  A  rectangular  piece  of  ground  contains  5  acres,  and 
the  width  is  20  rods.     Required  the  length. 

The  field  contains  800  square  rods,  and  we  wish  to  find  how 
many  rods  long  it  is.  If  it  were  only  ONE  rod  long  and  20  rods 
wide,  it  would  contain  20  square  rods.  But  as  it  contains  800 
rods,  it  must  be  8a°(j0  times  one  rod  long,  =  rods  ? 

20.  A  stick  of  hewn  timber  llin.  wide  and  lOin.  thick, 
contains  40  cubic  ft.     What  is  its  length  ? 


§59.]  MISCELLANEOUS   EXAMPLES.  195 

How  many  feet  long  ?  If  it  were  ONE  ft.  long,  y|ft.  wide,  and 
|ft.  thick,  it  would  contain  1  X  yj  X  |  —  f  |  cubic  ft.  But  as  it 
contains  40  c.  ft.  it  must  be  (40  -j-  f  f  )  times  ONE  ft.  long,  =  ft.  ? 

21.  If  A.  can  do  £  of  a  piece  of  work  in  5  days,  B.  can 
do  i  of  it  in  4  days,  and  C.  can  do  i  of  it  in  2  days,  in  how 
many  days  can  they  do  the  whole  by  working  together  ? 

In  ONE  day  they  can  do  y1^  +  y1^  +  y1^  of  the  1  piece  of  work. 
Then  they  can  do  the  1  piece  of  work  in  1  -j-  (y*o  +  yj  +  y^)  = 
days? 

59.    EXAMPLES  ILLUSTRATING  THE  SECOND  METHOD. 

1.  The  greater  of  two  numbers  is  5£  times  the  less,  and 
the  sum  of  the  numbers  is  52.     What  are  the  numbers  ? 

52  is  produced  by  adding  the  less  number  to  5£  times  the  less, 
and  is  therefore  6|  times  the  less. 

2.  Divide  the  number  582  into  four  such  parts  that  the 
second  may  be  twice  the  first,  the  third  21  more  than  the 
second,  and  the  fourth  54  more  than  the  first. 

582  is  produced  by  adding  21  and  54  to  6  times  the  first  num 
ber.  Therefore  if  we  subtract  75  from  582,  the  remainder  will 
be  6  times  the  first. 

3.  A  farmer  bought  some  horses,  cows,  and  calves  for 
$1250,  giving  $50  apiece  for  the  horses,  $23  apiece  for  the 
cows,  and  $9  apiece  for  the  calves,  and  there  were  three 
times  as  many  calves  as  cows,  and  half  as  many  horses  as 
calves.     How  many  were  there  of  each  ? 

If  there  had  been  2  cows,  6  calves,  and  3  horses,  they  would 
have  cost  $250.  But  he  paid  §1250,  then  how  many  times  could 
he  repeat  the  purchase  of  2  cows,  6  calves,  and  3  horses  ? 

4.  If  from  5  £  times  a  certain  number  8£  be  subtracted, 
12  i-  added  to  the  remainder,  and  the  sum  divided  by  6i, 
the  quotient  will  be  30.     What  is  the  number  ? 


5.  Five-eighths  of  a  certain  number  exceeds  $  of  it  by 
21.     What  is  the  number  ? 


196  GENERAL  ANALYSIS.        [ART.  XIV. 

.     If  21  is  24  X  the  number,  the  number  itself  is 


n+A- 

6.  What  number  is  that  from  which  if  we  deduct  |  of 
itself,  and  -f  of  the  remainder,  there  will  be  18  left  ? 

18  is  -J  of  what  is  left  after  deducting  f  of  the  number.  Then 
the  whole  number  is  |  of  \  of  18  =  ? 

7.  If  30  per  cent,  is  lost  by  selling  shoes  at  87 lets,  per 
pair,  at  what  price  should  they  be  sold  to  gain  10  per  cent.  ? 

If  30  per  cent,  is  lost,  they  must  be  sold  at  70  per  cent,  of  the 
cost.  To  gain  10  per  cent.,  they  should  be  sold  at  110  per  cent, 
of  the  cost,  i  LO  of  .87J=  ? 

8.  B.  is  2  years  older  than  A.,  C.'s  age  is  4  years  more 
than  the  sum  of  A.'s  and  B.'s,  and  D.'s  age,  which  is  48, 
is  equal  to  the  sum  of  the  other  three.     What  is  the  age 
of  each? 

48  is  the  sum  of  A.'s,  B.'s,  and  C.'s.  B.'s  is  2  more  than  A.'s, 
and  C.'s  is  6  more  than  twice  A.'s;  the  three  ages  are  therefore  8 
more  than  4  times  A.'s  age. 

6O.    MISCELLANEOUS  EXAMPLES  IN  ANALYSIS. 

1.  What  is  the  interest  of  $1872.88  for  7yr.  lOmo.  15dy. 
at  7  per  cent.  ?  Ans.  $1032.43. 

2.  Bought  18.75yd.   of  broadcloth,  at  $4  per  yard,  and 
sold  the  whole  for  $83.50.     What  did  I  gain,  and  how  much 
per  cent.  ?  Ans.  $8.50  =  Hi  per  cent. 

3.  At  what  rate  per  cent,  will  $421.50  amount  to  $674.40 
in  6yr.  8mo.  ? 

4.  A  note  of  $431  amounted,  at  its  settlement,  to  $546.29  J. 
How  long  had  it  been  on  interest,  the  rate  being  6  per  cent.  ? 

Ans.  4y.  5mo.  15dy. 

5.  What  is  the  face  of  a  note  which,  at  7  per  cent.,  will 
yield  $111.65  interest  in  3y.  8mo.  ?  Ans.  $435. 


§60.] 


MISCELLANEOUS    EXAMPLES. 


197 


6.  What  is  the  present  worth  of  $2000,  due  in  3y.  6mo., 
interest  at  7  per  cent.  ?  Ans.  $1606.43. 

7.  What  must  be  the  face  of  a  note  at  60  days,  to  be 
discounted  at  Bank  at  7  per  cent.,  to  yield  $375  ? 

Ans.   $379.65. 

8.  A  commission  merchant  received  2£  per  cent,  for  the 
sale  of  an  invoice  of  merchandise.     What  was  the  amount 
of  the  invoice,  the  total  amount  of  the  sale  and  commission 
being  $1666.24  ?  Ans.  $1625.60. 

9.  When  exchange  on  England  is  at  a  premium  of  91-  per 
cent.,  what  is  the  value  in  sterling  money,  of  $137.75  ? 

Ans.  28Z.  7s. 


10.  How  many  days  of  8£  hours,  will  42  men  require,  to 
build  a  wall  98fft.  long,  7*ft,  high,  and  2|ft.  thick,  if  63 
men  can  build  a  wall  45£ft.   long,  6T72ft.  high,  and  3$ft. 
thick,  in  68  days  of  Hi-  hours  ?  Ans.  297  days. 

11.  Determine  from  the  following  table,  the  degree  on 
each  thermometer-scale,  that  corresponds  to  0°  of  Fahren 
heit. 


Name  of  Thermometer. 

Where  used. 

Freezing  point- 

Boiling  point. 

Fahrenheit's     .     .     . 

Great  Britain  & 
United  States. 

4-3201 

1   212° 

Centigrade,  or  Celsius's 

Sweden  and 
France. 

0° 

+  100° 

Reaumur's  . 

jrermany,  Italy, 

0° 

I   80° 

Russian,  or  Delisle's 

Spain&  France. 
Russia. 

—  150° 

0° 

Ans.  Centigrade,  17|°  below  0. 
Reaumur's,  14  f°  "  0. 
Russian,  176§°  «  0. 

12.  How  many  square  yards  in  a  room  19ft.  6in.  wide, 
and  34ft.  8in.  long  ?  Ans.  75|  sq.  yd. 


a  The  degrees  above  zero  are  indicated  by  :hc  sign  -|-  ;  the  degrees 
below  zero  by  the  sign  — . 


198  GENERAL  ANALYSIS.  [ART.  XIV. 

13.  If  33  copecks  are  equal  to  5  English  pence,  11  Eng 
lish  pence  are  equal  to  3  piasters,  13  piasters  are  equal 
to    1  florin,  and    5  florins   are  equal   to   29  francs,  how 
many  francs  are  equal  to  11000  copecks  ? 

Ans.  202-J-J*  francs. 

14.  A.  contributed  $380  to  an  adventure,  B.  $420,  C. 
$500,  and  D.  $700.    What  was  each  man's  share  of  the  gain, 
which  was  $900  ?  Ans.  A.'s  share  $171 ;  B.'s  $189; 

C.'s$225;  D/s$315. 

15.  Divide  7500  into  5  parts,  in  the  proportions  of  £,  £, 
J,  i,  J-  A"*-  2586|f;  1724 Jf ;  1293&; 

1034ff;  862gV 

16.  An  echo  on  the  north  side  of  Shipley  Church,  in 
Sussex,  England,  repeats  21  syllables.*    If  the  speaker  utters 
3  syllables  a  second,  at  what  distance  from  the  echo  does 
he  stand,  the  velocity  of  sound  being  1090ft.  per  second? 

Ans.  3815ft. 

17.  A  man  failing  in  trade  owed  875000,  to  meet  which 
he  had  property  valued  at  $14500.     How  much  can  he  pay 
A.,  who  is  a  creditor  for  $10000,  B.,  who  is  a  creditor  for 
$3750,  and  C.,  who  is  a  creditor  for  $12362.50? 

Ans.  A.  $1933^;  B.  $725;  C.  $2390.08. 

18.  The  amount  contributed  by  the  United  States  in 
1847,  for  the  relief  of  Ireland  and  Scotland,  has  been  esti 
mated  at  $591313.29.b     In  what  time  would  10  men-of-war 
consume  the  same  amount,  supposing  each  vessel  to  have 
3  officers,  20  midshipmen,  and  1000  sailors;  estimating  the 
wages  and  rations  of  each  officer  at  $30,  of  each  midshipman 
at  $20,  and  of  each  sailor  at  $16  per  month  ? 

Ans.  3mo.  17.57  H-dy. 

19.  A  farmer  mixed  18J  bushels  of  wheat,  at  Sl.OO  per 
bushel;  16fbu.  at  $1.12i  per  bushel;  13fbu.  of  barley,  at 

»  Pierce.  b  Am.  Almanac,  1848. 


' 

§60.]  MISCELLANEOUS   EXAMP^ESf  199 

62|cts.  per  bushel,  and  lObu.  of  oats,  at  37Jcts.  per  bushel. 
What  was  the  mixture  worth  per  peck?     Ans.  $0.21  +  . 

20.  At  what  temperature  does  the  mercury  indicate  the 
same  degree  on  Fahrenheit's  and  on  the  Centigrade  scale  ? 
[See  Ex.  11.]  Am.  -  40°.a 

21.  Manson  &  Hill,  of  Liverpool,  have  given  to  Thomas 
Morton  &  Co.,  of  New  York,  a  note  of  £225  10s.,  payable 
in  60  days;  one  of  £196,  payable  in  60  days;  one  of  £218 
7s.  6d.,  payable  in  90  days ;  and  one  of  £300,  payable  in  120 
days.     At  what  time  may  the  notes  all  be  equitably  cancelled 
by  a  single  payment  of  £939  17s.  6d.  ?       -4ns.  86  days. 

22.  The  length  of  a  floor  is  37ft.  6in.,  and  the  area  is 
93f  sq.  yd.     What  is  the  width?  Ans.  22ft.  6in. 

23.  A  stick  of  hewn  timber  1ft.  2in.  wide  and  9in.  thick, 
contains  60f  solid  feet.     What  is  its  length  ? 

Ans.  69ft.  4iri. 

24.  A  grain  of  musk  is  said  to  be  capable  of  perfuming 
for  several  years,  a  chamber  12ft.  square,  without  sensible 
diminution  of  volume  or  weight. b     If  the  chamber  is  8ft. 
high,  and   constantly  contains  an  average  of  1  particle  to 
every  cubic  tenth  of  an  inch,  how  many  particles  must  there 
be  in  the  grain,  supposing  it  to  have  lost  J-Q^-Q  of  its  weight 
after  the  air  has  been  changed  5000  times  ? 

Ans.  9953280000000000. 

25.  A  father's  age  is  6|  times  his  son's  age,  and  the  sum 
of  their  ages  is  34yr.  2mo.   12dy.      What  is  the  age  of 
each? 

26.  Divide  $5000  into  four  such  parts,  that  the  first  may 
be  twice  the  second,  the  third  $50  less  than  £  the  second, 
and  the  fourth  $100  more  than  the  sum  of  the  first  and  third. 

27.  At  what  temperature/  does  the  mercury  indicate  the 

a  If  100°  gain  80°,  how  many  degrees  will  gain  32°  ? 
b  Moseley. 


200  GENERAL  ANALYSIS.        [ART.  XIV. 

same  degree  on  Fahrenheit's  and    on    Reaumur's    scale? 
[See  Ex.  11.]  Am.   —  25.6°. 

28.  A  farmer  hired  a  certain  number  of  boys,  and  twice 
as  many  men,  agreeing  to  pay  each  man  75  cents  a  day, 
and   each  boy  25  cents.     The  daily  wages  of  the  whole 
amounted  to  $5.25.     How  many  were  there  of  each  ? 

Ans.  3  boys;  6  men. 

29.  A  man's  age  is  such  that  if  it  be  multiplied  by  3, 
and  if  |  of  the  product  be  tripled,  |  of  the  result  will  be 
16.     Required  his  age.  Ans.  28. 

30.  What  is  that  number,  f  of  f  of  which  exceeds  It  of 

itself  by  21?  }*$ 

Ans.  147. 

31.  At  what  temperature  would  the  mercury  indicate  the 
same  degree  on  the  Centigrade  and  Russian  scales  ?     On  the 
Russian  and  German  scales  ?     [See  Ex.  11.] 

Ans.   +300°;   +  171f . 

32.  What  number  is  that,  from  which  if  we  deduct  f 
of  j|  of  itself,  and  f  of  J  of  the  remainder,  there  will  be 
10 J  left?  Ans.  34. 

33.  If  18  per  cent,  is  lost  by  selling  merchandise  at 
$2050,  at  what  price  should  it  have  been  sold  to  gain  10 
per  cent.  ? — to  gain  25  per  cent.  ? — to  lose  10  per  cent.  ? 

3d  Ans.  $2250. 

34.  B.  is  5  years  older  than  A. ;  C.'s  age  is  5  years  more 
than  the  sum  of  A.'s  and  B/s ;  and  D.'s  age,  which  is  55, 
is  equal  to  the  sum  of  the  other  three.     What  is  the  age  of 
each  ?  Ans.  A.  10  ;  B.  15  ;  C.  30. 

35.  In  examining  a  piece  of  charcoal  through  a  microscope, 
Dr.  Hook  counted  150  pores  in  Jg  of  an  inch.a     At  this 
rate  how  many  pores  would  there  be  in  one  square  inch  of 
surface?  Ans.  5760000. 

*  Moseley. 


§60.]  MISCELLANEOUS   EXAMPLES.  201 

36.  A.  and  B.  can  do  j  J  of  a  piece  of  work  in  1  day ;  B. 
and  C.  can  do  T9^  of  it;   A.  and   C.  can  do  |  of  it  in  the 
same  time.    In  what  time  will  they  all  do  it  working  together  ? 

By  adding  y-g>  y9g,  and  |,  and  dividing  the   sum  by  2,  we  find 
the  part  that  they  will  all  do  in  1  day. 

Ans.  |&  of  a  day. 

37.  Three  men  traded  in  partnership.     A.   contributed 
$1500,  B.  $2250,  and  C.  the  remainder.     The  whole  gain 
was  $2700,  of  which  C.  received  $1200.     How  much  did 
C.  contribute,  and  what  did  A.  and  B.  gain  ? 

Ans.  C.  contributed  $3000;  A.  gained  $600; 
B.  gained  $900. 

38.  An  estate  of  $15000  is  to  be  divided  among  three 
persons;  A.  is  to  receive  $5|-  as  often  as  B.  receives  $4£, 
and  B.  is  to  receive  $8?  as  often  as  C.  receives  $4^.     What 
is  the  share  of  each?         Ans.  A.  $6620.69;  B.  $5586.21; 

C.  $2793.10. 

39.  A  cistern  has  three  pipes;  the  first  can  fill  it  in  J  an 
hour,  the  second  can  fill  it  in  £  of  an  hour,  and  the  third 
can  empty  it  in  an  hour.     In  what  time  will  the  cistern  be 
filled,  if  they  all  run  together  ?  Ans.  15  minutes. 

40.  At  what  temperature  is  the  mercury  as  many  degrees 
above  zero  of  Fahrenheit,  as  it  is  below  zero  of  the  Centi 
grade?1     [See  Ex.  11.]  Ans.  llf. 

41.  A  bathing-tub  that  holds  147  gallons,  is  filled  by  a 
pipe  that  brings  14  gallons  in  9  minutes,  and  emptied  by  a 
pipe  that  discharges  40  gallons  in  30  minutes.     Both  pipes 
having  been  left  open  for  3  hours,  it  is  required  to  find  in 
what  time  the  tub  will  be  filled  if  the  discharging  pipe  is 
closed?  Ans.  Ih.  8T|  minutes. 

42.  What  sum  of  money  will  amount  to  $1500,   in  5 
years,  at  5  per  cent,  simple  interest?  Ans.  $1200. 

a  If  the  Centigrade  moves  100°  when  the  sum  of  the  motions  is  280°, 
how  much  will  it  move  when  the  sum  of  the  motions  is  32°  ? 


202  GENERAL  ANALYSIS.        [ART.  XIV 

43.  At  what  rate  per  cent.,  simple  interest,  will  $700 
amount  to  31300  in  11  years  ?  Ans.  7jJ^  per  cent. 

44.  In  what  time  will  $1100  amount  to  $1750,  at  6  per 
cent,  simple  interest?  Ans.  9||  years. 

45.  Supposing  the  weight  of  a  molecule  of  light  to  be 
roooouu  °f  a  grain)  what  should  be  the  velocity  of  a  ball, 
weighing  loz.  avoirdupois,  to  have  the  same  momentum.1 

Ans.  2.317+  ft.  per  second. 

46.  A  laborer  received  $1.50  for  every  day  he  worked, 
and  lost  50  cents  every  day  he  was  idle.     He  worked  twice 
as  many  days  as  he  was  idle,  and  at  the  end  of  the  time 
he  received  $42.50.     How  many  days  did  he  work? 

Ans.  34  days. 

47.  A  floor  is  laid  with  boards  l£in.  thick.     How  many 
feet  are  required,  the  room  being  18ft.  Gin.  wide,  and  20ft. 
Sin.  long?  Ans.  477j|ft. 

48.  How  many  clapboards  would  be  required  to  cover  an 
area  of  1376  sq.  ft.,  the  clapboards  being  4ft.  long,  and  laid 
with  4  inches  to  the  weather  ?  Ans.  1032. 

49.  A  cellar  is  to  be  made  40ft.  long,  and  25ft.  wide. 
How  many  squares  of  earth  must  be  removed,  the  depths 
at  six  different  points  being  8ft.,  7ft.  6in.,  7ft.  3in.,  8ft.  4in., 
8ft.  9in.,  and  5ft.  2in.  ?b  Ans.  34  j§  squares. 

50.  At  what  temperature  is  the  mercury  as  many  degrees 
above  zero  of  Fahrenheit,  as  it  is  below  zero  of  Reaumur  ? 
[See  Ex.  11.]  Ans.  9-J--J-0. 

51.  What  is  the  weight  of  a  stone  wall,  the  solid  con 
tents  being  2760  c.  ft.,  and  the  specific  gravity  2500? 

Ans.  431250  Ib. 

a  The  momentum  of  a  body  is  determined  by  multiplying  the  weight 
by  the  velocity.  The  velocity  of  light  is  192000  miles  per  second.  Such 
considerations  are  supposed  to  prove  that  light  is  without  weight. 

b  Assume  as  the  true  depth  of  the  cellar,  the  average  of  the  six 
measured  depths. 


§60.]  MISCELLANEOUS   EXAMPLES.  203 

52.  Wishing  to  estimate  the  contents  of  an  irregular  pile 
of  wood,  I  take  the  dimensions  in  several  places,  and  find  that 
the  average  length  is  34ft.  6in.,  the  average  breadth  27ft. 
4in.,  and  the  average  height  2ft.  Sin.     If  it  were  piled  regu 
larly,  I  judge  that  it  would  occupy  only  f  of  the  space  that 
it  now  does.     Required  its  estimated  contents. 

Ans.  11  cords  6£  c.  ft. 

53.  How  many  cubic  feet  in  a  stack  of  hay,  which  is 
estimated  to  be  equivalent  to  a  cylinder  12ft.  in  diameter 
and  15ft.  high  ?  Ans.  1696  c.  ft. 

54.  What  must  be  the  area  of  a  roof  that  would  fill  a 
cistern  holding  40  hogsheads,  with  a  fall  of  i  inch  of  rain, 
the  roof  being  of  a  true  pitch  ? 

The  roof  being  of  a  true  pitch,  the  area  of  the  roof  will  be  l-£ 
times  the  area  of  the  building,  but  the  water  that  falls  upon  it 
will  be  the  same  as  if  the  roof  were  flat. 

Ans.  24255  sq.  ft. 

55.  A  chronometer  usually  vibrates  4  times  in  a  second. 
How  much  must  the  length  of  each  vibration  be  increased, 
in  order  that  it  may  lose  1  second  per  day  ? 


56.  The  area  of  a  cistern  is  38  J  sq.  ft.     How  many 
gallons  would  fill  it  to  the  depth  of  1ft.  ? 

Ans.  288  gallons. 

57.  If  a  package  of  sugar  weighs  6  Ib.  2oz.  in  one  scale 
of  a  balance,  and  8  Ib.  in  the  other,  what  is  its  true  weight  ? 

Ans.  71b.a 

58.  A  man  performed  a  journey  of  135  miles,  going  twice 
as  far  the  second  day  as  on  the  first,  and  three  times  as  far  the 
third  day  as  on  the  second.     How  far  did  he  travel  each  day  ? 

a  The  true  weight  of  any  body  may  be  found  by  a  false  balance,  by 
weighing  the  body  in  each  scale,  and  taking  the  mean  proportional 
between  the  two  weights.  It  may  also  be  obtained  by  first  balancing 
the  body  with  shot  or  some  other  article,  and  then  removing  the  body 
and  placing  weights  in  the  scale  till  the  equilibrium  is  restored. 


204  GENERAL  ANALYSIS.        [ART.  XIV. 

59.  A.,  B.  and  C.  entered  into  partnership,  contributing 
in  the  whole,  $4833.     B.  paid  twice  as  much  as  A.,  and 
C.  paid  twice  as  much  as  A.  and  B.     How  much  did  each 
contribute  ? 

60.  In  a  certain  school  of  70  scholars,  three  times  as  many 
study  Arithmetic  as  study  Latin,  and  twice  as  many  learn 
to  read,  as  studj  Arithmetic.     How  many  are  there  in  each 
study  ? 

61.  What  degree  of  temperature  would  be  indicated  in 
the  same  manner  on  the  scales  of  Fahrenheit  and  Delisle  ? 
[See  Ex.  11.]  Ans.   -1060°. 

62.  An  estate  of  $7000  was  so  divided  that  the  widow 
received  $500  more  than  the  daughter,  and  the  son  $1100 
more  than  the  widow.     What  was  the  share  of  each  ? 

63.  Divide  the  number  97  into' four  such  parts  that  the 
second  may  be  twice  the  first,  the  third  7  more  than  the 
second,  and  the  fourth  18  more  than  the  first. 

64.  A  thief  travels  at  the  rate  of  6  miles  an  hour,  and 
after  he  has  been  absent  5£  hours,  a  constable  starts  in  pur 
suit,  at  the  rate  of  9  miles  an  hour.     In  what  time  will  the 
thief  be  overtaken  ? 

65.  A  man  when  he  was  married,  was  three  times  as  old 
as  his  wife,  but  after  they  had  lived  together  15  years,  he 
was  only  twice  as  old.     How  old  was  each  at  the  time  of 
marriage?  Ans.  45  years;  15  years. 

66.  At  a  certain  election,  the  successful  candidate  had 
163  votes  more  than  his  opponent,  and  the  whole  number 
of  votes  polled  was  1125.     How  many  did  each  receive  ? 

67.  What  sum  of  money  will  yield  $123.50  in  2  years, 
at  5  per  cent,  simple  interest  ? 

68.  A  gentleman  distributed  $1.95   among  3  beggars, 
giving  the  second  25  cents  more  than  the  first,  and  the 


§60.]  MISCELLANEOUS    EXAMPLES.  205 

third  twice  as  much  as  the  second.     How  much  did  each 
receive  ? 

69.  If  from  three  times  a  certain  number  17  be  subtracted, 
the  remainder  will  be  112.     What  is  the  number  ? 

70.  At  what  temperature  is  the  mercury  as  many  degrees 
below  zero  of  Delisle,  as  it  is  above  zero  of  Reaumur  ? — 
of  Celsius  ?— of  Fahrenheit  ?     [See  Ex.  11.] 

Ans.  52-2430;  60°;  96T4T°. 

71.  A  merchant  owes  two  of  his  creditors  $1575,  and  he 
owes  the  second  but  f  as  much  as  the  first.     "What  is  the 
amount  of  each  debt  ? 

72.  In  a  certain  school  J  *lie  boys  learn  to  read,  J  learn 
to  write,  y^  learn  Algebra,  325  learn  drawing,  and  the  remain 
ing  4  study  Latin.     How  many  are  there  in  the  school  ? 

73.  One-third  of  a  certain  pole  is  painted  green,  and  % 
of  it  is  painted  white,  the  remainder,  which  is  8  feet,  being 
in  the  ground.     What  is  the  length  of  the  pole  ? 

74.  A  man  going  to  market,  was  met  by  another,  who 
said  :  "  Good  morrow,  neighbor,  with  your  hundred  geese." 
He  replied :  "  I  have  not  a  hundred ;  but  if  I  had  as  many 
more,  and  half  as  many  more,  and  two  geese  and  a  half 
besides,  I  should  have  a  hundred."     How  many  had  he  ? 

75.  A  man  bought  38  pounds  of  coffee  and  95  pounds  of 
sugar;  he  gave  2  cents  per  Ib.  more  for  the  coffee  than  for 
the  sugar,  and  the  sugar  cost  twice  as  much  as  the  coffee. 
What  was  the  price  of  each  per  pound  ? 

Ans.  Sugar  Sets. ;  coffee  lOcts. 

76.  If  4  men  can  saw  15  cords  of  oak  in  the  same  time 
that  5  men  saw  14  cords  of  hickory,  and  if  3  men  saw  18 
cords  of  hickory  in  3  days,  by  working  9  hours  a  day,  how 
many  hours  a  day  must  7  men  work,  to  saw  84  cords  of  oak 
in  6  days?  Ans.  6^|  hours. 

77.  There  are  two  such  numbers,  that  if  21  be  added  to 


206  GENERAL  ANALYSIS.        [ART.  XIV. 

the  first,  the  sum  will  be  5  times  the  second,  and  if  21  be 
added  to  the  second,  the  sum  will  be  3  times  the  first. 
What  are  the  numbers  ? 

25  1  is  the  difference  between  I  of  the  first,  and  3  times  the 
first.  The  first,  therefore,  is  9,  and  the  second  6. 

78.  Two  stages  are  travelling  towards  each  other,  one  at 
the  rate  of  5|  miles  an  hour,  the  other  6  J  miles  an  hour.  In 
what  time  will  they  meet  if  they  are  now  38f  miles  apart? 

Ans. 


79.  Two  men  start  from  the  same  place  and  travel  in 
opposite  directions,  one  at  the  rate  of  4-J-  miles  an  hour,  and 
the  other  5  j3f  miles  an  hour.     In  what  time  will  they  be  100 
uiiles  apart?  Ans.   10]  §  hours. 

80.  With  what  velocity  must  a  battering  ram,  weighing 
2000  lb.,  be  moved,  to  have  the  same  momentum  as  a  cannon 
ball,  weighing  20  lb.  and  moving  1200ft.  per  second  ? 

Ans.  12ft.  per  second. 

81.  There  is  a  number  to  which  if  *,  |,  J,  and  |  of  itself 
be  added,  the  sum  will  be  f  of  8J  less  than  16.     What  is 
the  number?  Ans. 


82.  If  A.  can  do  £  of  a  piece  of  work  in  5  days,  B.  can 
do  £  of  it  in  4  days,  and  C.  can  do  -J-  of  it  in  2  days,  in 
what  time  will  they  all  do  f  of  it  by  working  together  ? 

Ans.  2  £  days. 

83.  There  are  two  men  of  equal  ages,  but  if  one  was  5J 
years  older,  and  the  other  9|-  years  younger,  the  former 
would  be  twice  as  old  as  the  latter.     Kequired  their  ages. 

Ans.  24yr. 

84.  If  a  man  can  do  T5T  of  a  piece  of  work  in  3  days,  and 
a  boy  can  do  |  of  it  in  5  days,  how  long  will  it  take  them 
both  to  do  the  whole  ?  Ans.  425T9?  days. 


85.  A  hare  starts  5  rods  before  a  greyhound,  and  runs  at 
the  rate  of  12  miles  an  hour.     After  running  48  seconds, 


§60.]  MISCELLANEOUS   EXAMPLES.  207 

the  hound  starts  in  pursuit,  and  runs  20  miles  an  hour.     In 
what  time  will  the  hare  be  overtaken  ? 

Arts.  1m.  lOg1^  sec. 

86.  If  300  tiles  that  are  9in.  long  and  6in.  wide,  will 
pave  a  court-yard,  how  many  tiles  would  be  required  that 
are  6in.  long  arid  4in.  wide  ?  Ans.  675  tiles. 

87.  How  many  men  will  build  a  wall  240yd.  long,  6ft. 
high,  and  3ft.  thick,  in  8  days  of  9  hours,  if  7  men  can 
build  a  wall  40yd.  long,  4ft.  high,  and  2ft.  thick,  in  32  days 
of  7  hours  ?  Ans.  294  men. 

88.  A  wall  which  is  to  be  built  to  the  height  of  27  feet, 
has  been  raised  9  feet  in  6  days,  by  12  men  working  13 
hours  a  day.     How  many  men  must  be  employed  to  finish 
it  in  2  days,  working  only  12  hours  a  day  ? 

Ans.  78  men. 

89.  Amsterdam  exchanges  with  London,  at  34  schillings 
4  pfennings  per  <£,  and  with  Lisbon  at  56  pfennings  for 
400  reas.     What  is  the  arbitrated  exchange  between  Lon 
don  and  Lisbon,  by  way  of  Amsterdam  ? 

Ans.  £l  =  2e942f. 

90.  Find  the  number  of  vibrations  in  one  second,  of  each 
of  the  rays  of  light,  the  velocity  of  light  being  192000  miles 
per  second,1  and  the  lengths  of  the  waves  being,  for  red  light, 
.0000256  of  an  inch. 


For  blue  .  .  .0000196 
"  indigo  .  .  .0000185 
"  violet  .0000174" 


For  orange  .  .  .0000240 
"  yellow  .  .  .0000227 
"  green  .  .  .0000211 

Ans.  Red,  475200000000000  vibrations. 
Violet,  699144827586207        « 
&c.         &c.  &c. 

91.  The  force  available  for  mechanical  purposes  in  an  adult 
man,  is  reckoned,  in  mechanics,  equal  to  *  of  his  own  weight, 

a  Herschel.  b  Draper. 


208  THE   COUNTING-HOUSE.  [ART.  XV. 

which  he  can  move  during  8  hours,  with  a  velocity  of  2  Jft. 
per  second.1  What  is  the  momentum5  for  the  day's  work 
of  a  man  who  weighs  160  Ib.  ?  Ans.  2304000. 


XV.    THE  COUNTING-HOUSE. 

6 1 .    PERCENTAGE. 

1.  THE  term  per  cent,  is  an  abbreviation  of  the  Latin  per 
centum,  which  signifies  l>y  the  hundred.     Any  number  of 
per  cent,  of  a  quantity  is  therefore  equivalent  to  as  many 
hundredths  of  that  quantity.     Thus  7  per  cent,  is  .07;  4£ 
per  cent,  is  .04  J  or  .045;  18f  per  cent,  is  .18f  or  .1875. 

2.  Per  cent,  should  not  be  confounded  with  any  of  the 
denominations  of  Federal  Money.     Thus,  6  per  cent,  is  not 

6  cents,  or  6  dollars,  but  simply  T|JQ.     6  per  cent,  of  125 
dollars,  is  ^  of  $125  =  87.50;  but  6  per  cent,  of  125 
apples  is  7.50,  or  7z  apples,  and  6  per  cent,  of  125  Ib.  is 

7  Ib.  8oz. 

3.  Any  fraction  may  be  reduced  to  per  cent.,  either  by 
reducing  it  to  a  decimal,  and  stopping  the  decimal  at  hun 
dredths'  place,  or  by  multiplying  the  fraction  by  100.    Thus 
T75  reduced  to  a  decimal,  gives  .46f,  or  46f  per  cent. ;  the 
same  fraction  reduced  to  hundredths  by  multiplying  by  100, 
gives  7j0-°  hundredths,  or  7T°5°  per  cent.,  or  46f  per  cent. 

4.  To  determine  the  value  of  any  quantity  when  there  is 
a  specified  gain  or  loss  per  cent.,  we  may  add  or  subtract 
the  given  percentage  from  jgg  or  1.     Thus,  if  stock  is  sold 


a  Liebig. 

b  The  momentum  is  obtained  by  multiplying  the  weight  by  the 
distance. 


§62.]  PROBLEMS    IN    PERCENTAGE.  209 

at  25  per  cent,  advance,  it  is  sold  for  |§J,  or  1.25  times  its 
par  value  ;  if  goods  are  sold  at  a  loss  of  18  per  cent.,  they 
are  sold  for  •  of  their  cost. 


.    PROBLEMS  IN  PERCENTAGE. 
I.    To  find  the  gain  or  loss  per  cent. 
Make  the  gain  or  loss  the  numerator,  and  the  prime  cost 
the  denominator  of  a  fraction,  and  multiply  the  resulting 
fraction  by  100.     [See  Sect.  61,  3.] 

a.  To  find  what  percentage  must  be  gained  on  the  selling  price, 
to  yield  any  desired  profit  per  cent,  on  the  cost. 

Divide  the  desired  percentage  of  gain,  by  100  plus  that  per 
centage,  and  reduce  the  quotient  to  hundredths. 

b.  To  find  the  percentage  of  profit  on  the  cost  of  merchandise, 
the  percentage  gained  on  the  selling  price  being  known. 

Divide  the  percentage  gained  on  the  selling  price  by  100  minus 
that  percentage,  and  reduce  the  quotient  to  hundredths. 

II.    To  determine  the  value  of  a  quantity,  when  the 
value  of  any  percentage  is  knoicn. 

Divide  by  the  percentage  expressed  decimally. 

EXAMPLE.  —  If  2500  is  16  per  cent,  of  a  certain  number,  what 
is  that  number  ?  Since  .16  of  a  number  =  .16  times  that  number, 
2500  must  be  .16  times  the  number  sought.  The  answer,  there 
fore,  is  2500  -=-.16  =  15625. 

a.  To  find  the  amount  which  should  be  added  to  an  insurance, 
to  recover  the  amount  of  premium  paid. 

Divide  the  premium  by  the  difference  between  the  rate  and  100 
per  cent. 

III.    To  find  the  result  that  any  quantity  will  yield  in 
percentage. 

Multiply  the  quantity  by  the  result  that  ONE  will  yield. 

a.  To  find  the  interest,  or  the  amount,  of  any  principal  for  any 
given  time. 

Find  the  interest,  or  the  amount,  of  ONE  dollar  for  the  given 
time,  and  multiply  by  the  NUMBER  of  dollars  in  the  given  princi 
pal.     This  rule  will  hold  good,  both  for  simple  and  compound 
interest. 
14 


210  THE   COUNTING-HOUSE.  [ART.  XV. 

To  find  the  interest  of  $1  for  any  given  time  :  To  J  as  many 
cents  as  there  are  months,  add  ^  as  many  mills  as  there  are 
remaining  days,  and  the  amount  will  be  the  interest  at  6  per 
cent.  For  any  other  rate,  take  such  part  of  the  interest  at  6  per 
cent,  as  may  be  requisite. 

EXAMPLE.  —  What  is  the  interest  of  $287.75  for  3y.  7rno.  23dy., 
at  5|  per  cent.  ?  The  interest  of  $1  at  6  per  cent.  =  £  of  43cts. 
-f-  J  of  23  mills,  or  $.218|.  At  5|  per  cent,  it  will  be  5£  sixths, 
or  -J-J  as  much,  or  ]4  X  ^l1-3-  The  interest  of  $287.75  = 
$287.75  x  !i  X  HL3  =  $57-72. 

b.  To  find  the  selling  price,  to  make  any  proposed  gain  or  loss 
per  cent. 

Multiply  the  prime  cost  by  1  with  the  percentage  added  which 
is  to  be  gained,  or  the  percentage  subtracted  which  is  to  be  lost. 

c.  To  reduce  Sterling  to  Federal  Money. 

Multiply  the  par  value  of  £1,  ($4-£,)  by  1  +  the  premium.  The 
product  will  be  the  exchange  value  of  £1. 

Multiply  the  exchange  value  of  £1,  by  the  number  of  pounds. 

IV.    To  find  the  quantity  that  will  yield  any  given 

result  in  percentage. 

Divide  by  the  result  that  ONE  would  yield. 
EXAMPLE.  —  What  principal  will  amount  to  $500,  in  2y.  6mo., 
at  5  per  cent.  ?     $1  would  amount  to  $1.125,  and  500  -^  1.125  = 


a.  To  reduce  Federal  to  Sterling  Money. 
Divide  by  the  exchange  value  of  £1. 

b.  To  find  the  face  of  a  note  to  be  discounted  at  bank,  in  order 
to  obtain  any  required  sum. 

Divide  the  sum  required,  by  the  amount  that  would  be  received 
by  discounting  $1.  The  quotient  will  be  the  number  of  dollars 
for  which  the  note  should  be  drawn. 

c.  To  find  the  amount  that  a  factor  can  lay  out  of  a  sum  intrusted 
to  him,  and  reserve  a  specified  percentage  for  his  commission. 

Divide  the  sum  intrusted  to  him,  by  1  +  the  percentage  which 
is  allowed  for  his  commission. 

d.  To  find  the  prime  cost,  when  the  selling  price  and  the  gain 
or  loss  per  cent,  are  known. 

Divide  the  selling  price  by  the  value  of  1  with  the  proposed  gain 
or  loss  per  cent. 


§63.]  EXAMPLES   IN   PERCENTAGE.  211 

e.  To  find  the  selling  price  so  as  to  allow  a  discount  for  cash, 
and  gain  any  proposed  rate  per  cent. 

Multiply  the  prime  cost  by  1  -|-  the  proposed  gain  per  cent., 
and  divide  by  1  —  the  proposed  discount  per  cent. 

/.  The  gain  or  loss  per  cent,  at  any  given  price  being  known,  to 
find  the  gain  or  loss  per  cent  at  any  proposed  price. 

Multiply  the  percentage  of  the  prime  cost  which  corresponds 
to  the  given  price,  by  the  proposed  price,  and  divide  by  the  given 
price.  The  quotient,  (reduced  to  hundredth s,)  will  be  the  percent 
age  of  the  prime  cost  which  corresponds  to  the  proposed  price. 
The  difference  between  this  percentage  and  100  per  cent.,  will  be 
the  gain  or  loss  per  cent. 

63.    EXAMPLES  IN  PERCENTAGE. 

A.  Examples  illustrating  \  61. 

1.  Find  25  per  cent,  of  $13.50;  7£  per  cent,  of  2cwt. 
3qr.  12  Ib. ;  16f  per  cent,  of  252  miles ;  4f  per  cent,  of 
£60;  135  per  cent,  of  lOmo. 

2.  How  many  per  cent,  are  equivalent  to  |;  to  | ;  |;  -J-|; 

^  .     8  .     37  .     1  a •        4      .    278  ? 
4  )    ?  >    ~8G  >    20>    55TJ  J     T3    { 

3.  What  part  of  the  original  value  is  stock  worth,  when 
it  is  15  J  per  cent,  below  par  ?     When  it  is  at  a  premium  of 
8f  per  cent.  ?     When  it  is  at  a  discount  of  12  £  per  cent.  ? 

4.  How  much  was  received  for  a  farm,  bought  for  $1350, 
and  sold  at  an  advance  of  33i  per  cent.  ?      Ans.  $1800. 

B.  Examples  under  Problem  I.  g  62. 

5.  How  much  per  cent,  was  lost  on  flour,  which  was 
bought  at  $5,  and  sold  at  $4.62 J  per  barrel? 

Ans.  7£  per  cent. 

6.  What  percentage  must  a  merchant  gain  on  the  total 
amount  of  his  sales,  to  be  equivalent  to  a  gain  of  10  per 
cent,  on  the  cost  ?  Ans.  9^  per  cent. 

7.  A  tradesman  finds  that  his  profits  on  a  year's  business 
amount  to  1 6|-  per  cent,  of  the  sales.     What  percentage  of 
the  cost  has  he  gained?  Ans.  20  per  cent 


212  THE   COUNTING-HOUSE.  [ART.  XV. 

C.  Examples  under  Problem  II.  g  62. 

8.  Twenty-seven  and  a  half  is  18  per  cent,  of  what  num 
ber?  Am.  152*. 

9.  What  sum  should  be  insured  to  cover  the  amount 
paid  for  premium  and  policy,  if  I  wish  to  insure  $1800  on 
merchandise,  at  a  premium  of  f  per  cent.,  the  charge  for 
the  policy  being  $1  ?  Ans.  $1814.61. 

D.  Examples  under  Problem  III.  §  62. 

10.  Find  the  interest  of  £27  7s.  6d.  for  ly.  7mo.  18dy. 
at  5  per  cent.  Ans.  £2  4s.  8|d. 

11.  At  what  price  should  I  sell  broadcloth,  which  cost 
$3 1  per  yd.,  in  order  to  gain  11 J  per  cent.  ? 

Ans.  $3.75. 

12.  At  9&  per  cent,  premium,  what  is  the  value  in  U.  S. 
currency,  of  £27  13s.  ?  Ans.  $134.56. 

E.  Examples  under  Problem  IV.  §  62. 

13.  What  principal,  at  7  per  cent,  compound  interest, 
would  yield  an  interest  of  $500  every  third  year  ? 

Ans.  $2221.80. 

14.  At  9  per  cent,  premium,  what  will  be  the  value  in 
English  money,  of  a  Bill  of  Exchange  for  $1250.25  ? 

Am.  £258  Is.  7d. 

15.  Required  the  face  of  a  note  at  90  days,  to  yield  $500 
when  discounted  at  bank.  Ans.  $507.87. 

16.  A   factor  receives  5    per  cent,  commission  on  the 
amount  that  he  purchases.     If  I  send  him  $1000,  how  much 
can  he  lay  out,  after  reserving  enough  to  pay  his  own  com 
mission?  Ans.  $952.38. 

17.  If  I  gain  15  per  cent,  by  selling  land  at  $402.50  per 
acre,  what  did  the  land  cost  per  acre  ?  Ans.  $350. 

18.  At  what  price  must  I  sell  molasses  that  cost  25cts. 
per  gallon,  in  order  to  gain  10  per  cent.,  after  discounting 
5  per  cent,  for  cash  ?  Ans.  $.28  jf  per  gallon. 


§64.]  PERCENTAGE   ON   STERLING  MONEY.  213 

19.  If  12  £  per  cent,  is  gained  by  selling  a  house  foi 
$3825,  what  percentage  would  be  gained  or  lost  by  selling 
it  for  $3230  ?  Ans.  5  per  cent.  lost. 

F.  Miscellaneous  Examples. 

20.  A  bill  of  grods  is  purchased  on  6  months  credit,  amount 
ing  to  $175.75.     How  »>uch  should  be  paid  in  cash,  at  the 
time  of  purchase,  if  the  buye?  is  allowed  5  per  cent,  for 
his  money  ?  Ans.  $167.38. 

21.  At  what  rate,  simple  interest,  will  any  principal  be 
doubled  in  12y.  6mo.  ? 

22.  In  what  time  will  $2700  amoun*  to  $3132,  at  6  per 
cent,  simple  interest  ? 

23.  What  sum  of  money  at  6  per  cent,  compound  interest, 
will  amount  to  $2750,  in  3y.  6mo.  ?         Ans,  $2241.70. 

24.  If  a  merchant  receives  his  usual  profii,  by  selling  a 
quantity  of  sugar  for  £46  5s.,  how  much  must  he  raise  the 
price,  in  order  to  allow  a  discount  of  7£  per  cent.  ? 

Ans.  £3  15s. 


PERCENTAGE  ON  STERLING  MONEY. 

In  computing  interest  or  discount  on  English  money, 
for  any  length  of  time  less  than  a  year,  it  is  customary  to 
omit  the  shillings  and  pence  in  the  principal,  when  they  are 
less  than  ten  shillings  ;  but  if  they  amount  to  ten  shillings 
or  more,  they  are  considered  as  another  pound. 

If  it  is  desired  to  compute  the  percentage  exactly,  it  may 
be  done  either  by  reducing  the  shillings  and  pence  to  the 
fraction  or  decimal  of  a  pound,a  or  by  multiplying  1  per 

a  Shillings,  pence,  and  farthings  may  be  reduced  to  the  decimal  of  a 
pound  by  inspection,  as  follows  :  —  Multiply  the  number  of  shillings  by 
5,  and  call  the  product  hundredths.  Reduce  the  pence  and  farthings  to 
farthings,  increasing  their  number  by  1,  when  it  exceeds  12,  and  by  2, 
when  it  exceeds  36,  and  call  the  result  thousandths.  The  sum  of  these 
two  values  will  be  the  decimal  required. 

To  reduce  the  decimal  of  a  pound  to  shillings,  pence  and  farthings, 


214 


THE   COUNTING-HOUSE. 


[ART.  XV. 


cent,  of  each  denomination,  by  the  number  of  per  cent, 
required.     The  latter  method  is  generally  the  readiest. 

EXAMPLE   ILLUSTRATING   EACH   METHOD. 

Find  3£  per  cent,  of  £480  10s.  3d. 


First  Method. 
10s.  3d.  =  .5125£ 

480.5125 
.035 


24025625 
14415375 

16.8179375 
20 


16.3587500 
12 

4.30500 
4 


1.220 
An*.  £16  16s.  4R 


Second  Method. 

s.  d. 

1  per  cent.  =  £4.80  .10  .03 

3>=  7 


2)33.60  .70  .21 

16.80   .35  .105 
20 


1.220 
Ans.  £16  16s.  4id. 


The  legal  rate  of  interest  in  England  is  5  per  cent.  For 
computing  interest  at  this  rate,  the  following  rules  are  con 
venient.  In  each  case,  the  principal  is  supposed  to  be 
expressed  in  pounds,  and  parts  of  a  pound. 

1.  Multiply  the  principal  by  the  number  of  years,  and 
the  product  will  be  the  interest  in  shillings. 

2.  Multiply  the  principal  by  the  number  of  months,  and 
the  product  will  be  the  interest  in  pence. 

3.  Multiply  the  principal  by  the  number  of  days,  and  divide 
the  product  by  30;  the  quotient  will  be  the  interest  in  pence. 

For  any  other  rate  than  5  per  cent.,  first  compute  the 

Multiply  the  number  of  tenths  by  2,  and  the  product  will  be  shillings. 
From  the  remainder  of  the  decimal  subtract  Jj  of  itself ,  and  the  figures 
which  stand  in  the  hundredth*'  and  thousandths'  places  will  be  farthings. 


§65.]  BANKING.  2l£> 

interest  at  5  per  cent.,  and  multiply  the  result  by  .2  X  the 
number  of  per  cent.  For  3  per  cent.,  multiply  by  .6 ;  for 
4£  per  cent.,  multiply  by  .9;  &c.  &c. 

EXAMPLES. 

1.  What  is  the  interest  of  £487  10s.  8d.,  from  March 
4th,  to  Dec.  17,  at  6  per  cent.  ? 

Mercantile  Ans.  £23  Os.  4£d. 
Correct  Ans.  £22  19s.  lid. 

2.  What  per  cent,  is  gained  by  selling,  at  £1  10s.,  velvet 
that  cost  £1  2s.  6d.,  per  yard  ?  Ans.  33£  per  cent. 

3.  How  much  was  received  for  25  shares  of  stock,  sold 
at  a  premium  of  8|  per  cent.,  the  par  value  being  £50  per 
share  ?  Ans.  £1359  7s.  6d. 

4.  What  was  the  interest  of  six  India  Bonds,  of  £100 
each,  at  3£  per  cent.,  calculating  from  Sept.  30,  1849;  the 
Bonds  having  been  sold  Jan.  15,  1850  ? 

Ans.  £6  2s.  Gd. 

5.  What  amount  of  4  per  cent,  stock,  will  yield  an  income 
of  £150  per  annum  ?  Ans.  £3750. 

Gt>.    BANKING. 

In  computing  interest  at  Bank,  the  time  is  usually  deter 
mined  in  days.  When  the  rate  is  6  per  cent.,  the  interest 
is  found  by  multiplying  the  principal  by  as  many  thousandths 
as  are  equivalent  to  the  number  of  days,  and  dividing  the 
product  by  6.  Thus  the  Bank  interest  of  $175.50,  for  63 
days,  is  $175.50  X  .063  -=-  6  =  $1.84. 

For  any  other  rate  than  6  per  cent.,  we  may  first  compute, 
the  interest  at  6  per  cent.,  and  add  or  subtract  such  part  as 
may  be  required.  For  4  per  cent.,  subtract  £  of  the  interest 
at  6  per  cent.;  for  4J  per  cent,  subtract  J;  for  5  per  cenfc. 
subtract  i ;  for  7  per  cent,  add  i,  and  so  on. 


216  THE   COUNTING-HOUSE.  [ART.  XV, 

If  a  note  is  given,  or  a  bill  drawn  for  any  number  of 
months,  calendar  months  are  always  understood.  A  note 
at  4  months,  dated  on  the  29th,  30th,  or  31st  of  October, 
would  expire  on  the  last  day  of  February,  and  would  be 
legally  due  on  the  3d  of  March.  The  3d  of  March  is,  there 
fore,  a  heavy  day  at  bank,  as  in  leap  years  there  are  3  days' 
payments,  and  in  common  years  4  days'  payments,  which 
fall  due  on  that  day.  If  either  the  3d  or  4th  of  March,  in 
any  year  except  leap  year,  falls  upon  Sunday,  there  will  be 
5  days'  payments  falling  due  on  the  Saturday  previous. 

There  are  two  modes  of  estimating  the  time  that  elapses 
between  different  dates. .  The  first,  is  by  compound  sub 
traction,  which  is  the  method  almost  invariably  adopted  in 
computing  interest  on  notes,  payable  on  demand.  The 
second,  is  by  determining  the  number  of  entire  calendar 
months,  and  then  finding  how  many  days  are  left.  This 
mode  is  adopted  in  many  counting-houses,  and  in  all  banks. 
Thus,  from  Oct.  27th,  1850,  to  March  15th,  1853,  would 
be,  according  to  the  1st  method,  2y.  4mo.  18dy. 
"  "  "  2d  "  2y.  4mo.  16dy. 

From  Oct.  31st,  1850,  to  March  15th,  1853,  would  be, 
according  to  the  1st  method,  2y.  4mo.  14dy. 
"  "     "  2d       «         2y.  4mo.  15dy. 

Bank  discount  is  the  same  as  Bank  interest.  If  a  note 
is  discounted  at  bank,  the  bank  takes  off  the  interest  for  the 
time  the  note  has  to  run,  and  pays  the  balance  only  to  the 
holder  of  the  note. 

The  number  of  days  which  elapse  between  two  given 
dates,  may  be  found  as  in  the  following  example : 

Required  the  number  of  days  between  March  23d,  and 
Sept.  5th. 

We  find  by  adding  the  number  of  days  in  all  the  intervening 
time,  that  Sept.  5th  would  correspond  to  March  189th,  if  the 
days  were  all  regarded  as  belonging  to  March.  Between  March 


§65.] 


BANKING. 


217 


23d,  and  March  189th,  would  be  166  days,  or     March  has  31  days 
23  weeks  and  5  days.a 

If  we  wish  to  find  on  what  day  of  the  week 
any  given  date  will  fall,  we  may  proceed  in 
a  similar  manner.  Thus,  if  March  23d  comes 
on  Friday,  as  there  are  23  complete  weeks, 
and  5  additional  days  between  March  23d  and 
Sept.  5th,  Sept.  5th  will  fall  5  days  after 
Friday,  or  on  Wednesday. 


April 

May 

June 

July 

August 


30 
31 
30 
31 
31 


Add  for  Sept.  5 

~189~ 
23 


166  dy. 

A  note  at  -30  days,  will  fall  due  in  33  days=4wk.  5d. ;  a  note 
at  60  days,  in  63  days,  =9wk.  ;  a  note  at  90  days,  in  93  days,  = 
13wk.  2d. ;  a  note  at  120  days,  in  123  days,  =  17wk.  4d.  There 
fore  a  30  days'  note  becomes  due  5  week  days  later  than  the  day 
on  which  it  is  given ;  a  60  days'  note,  on  the  same  day  of  the  week 
as  the  day  on  which  it  is  given ;  a  90  days'  note,  2  week  days  later  ; 
a  120  days'  note,  4  week  days  later.  It  is  sometimes  important  to 
date  a  note  so  that  it  will  not  mature  on  Sunday,  or  on  a  holi 
day.  This  can  easily  be  done  by  one  of  the  methods  above  given. 


EXAMPLES. 

1.  How  much  would  an  English  merchant  receive  on  a 
note  for  £600  at  4  months,  discounted  at  4  per  cent.  ? 

Ans.  £591  16s. 

2.  When  will  a  3  months'  note  fall  due,  if  dated  Aug.  31  ? 
— a  note  at  8  months,  dated  June  30th  ? 

a  The  number  of  days  may  also  be  found  by  the  following  table,  when 
the  time  is  less  than  a  year  : 

TABLE  FOR  ASCERTAINING  THE  NUMBER  OF  DAYS  FROM  ANT  DAY  m 

THE    YEAR,    TO    ANY   OTHER   DAY. 


1st  Mo.,  Jan. 
2d  Mo.,  Feb. 
3d  Mo.,  Mar. 
4th  Mo.,  Apl. 


5th  Mo.,  May 
6th  Mo.,  June 
7th  Mo.,  July 
8th  Mo.,  Aug. 


120 
151 
181 
212 


9th  Mo.,  Sept.  243 

10th  Mo.,  Oct.  .  273 

llth  Mo.,  Nov.  304 

12th  Mo.,  Dec.  .  334 


RULE. 

To  the  given  day  of  each  month,  add  the  tabular  number  for  the  month, 
and  subtract  the  less  sum  from  the  greater. 

If  the  two  dates  are  in  different  years,  subtract  the  result  thus  found 
from  365. 

In  leap  years,  add  1  to  the  number  after  the  28th  of  February. 


218  THE    COUNTING-HOUSE.  [ART.  XV. 

3.  Determine   by  each  method,  the   time  that   elapsed 
between  July  4th,  1776,  and  June  2d;  1850;  between  Aug. 
18,  1820,  and  Feb.  29,  1848. 

4.  How  many  days  were  there  between- Jan.  27,  and  June 
16,  1844  ?— between  Feb.  20,  and  Oct  8,  1849  ? 

5.  New  Year's  day,  A.  D.  1850,  fell  on  Tuesday.     On 
what  day  of  the  week  was  Christmas  of  the  same  year  ? 

6.  Required  the  avails*  of  a  note  at  4  months,  for  $275.50, 
dated  Feb.  12th,  1850,  and  discounted  at  a  bank  in  New 
York,  where  the  legal  rate  of  interest  is  7  per  cent. 

Am.  $268.91. 

GG.    PARTIAL  PAYMENTS. 

When  partial  payments  are  made  on  mercantile  accounts 
which  are  past  due,  it  is  customary  to  compute  interest  on 
the  whole  debt  from  the  time  it  became  due,  and  on  each 
payment  from  the  time  it  was  made,  until  the  time  of  settle 
ment,  and  to  deduct  the  amount  of  all  the  payments,  including 
interest,  from  the  amount  of  the  debt  and  interest. 

The  labor  of  computing  interest  on  each  item  may  be 
avoided,  by  multiplying  the  amount  due  at  first,  and  the  bal 
ance  due  after  each  payment,  by  the  number  of  days  that 
they  are  severally  at  interest,  adding  all  the  products,  and 
dividing  the  amount  by  6000.  The  quotient  will  be  the 
interest  at  6  per  cent.b 

EXAMPLE  FOR  ILLUSTRATION. 

A  debt  of  $630.25  became  due  March  15th,  on  which  the  fol 
lowing  payments  were  made:  April  3d,  $170  ;  May  20,  $245.30; 
June  17th,  $87.50.  How  much  was  due  Sept.  5,  when  the  account 
was  settled? 

1  The  amount  received  for  the  note  after  it  is  discounted. 

b  It  may  be  readily  seen  that  the  same  result  will  be  obtained  which 
ever  method  is  adopted,  but  no  accountant  who  is  familiar  with  both 
methods,  will  hesitate  to  adopt  the  latter. 


§67.]  LEGAL   INTEREST.  219 

$        Days.    Products. 

March  15,  Amount  due       ...     630.25  X  19  =  11974.75 
April  3,  1st  payment  .         .         .         .170. 

Balance 460.25"  X  47  =  21631.75 

May  20,  2d  payment  ....     245.30 

Balance 2T4795~x28  =    6018.60 

June  17,  3d  payment  .         .         .         .       87.50 

Balance 127.45"  X  80=10196.00 

6000)  49821.10 

8.30351 

Ans.  $127.45+ $8.30  =  $135.75. 

The  whole  debt  is  on  interest  from  March  15  to  April  3,  19  days. 
The  principal  is  then  reduced  by  a  payment,  to  $460.25,  which  is 
on  interest  from  April  3  to  May  20,  47  days.  In  like  manner  we 
find  that  the  2d  balance  is  on  interest  28  days,  and  the  3d,  80  days. 
The  amount  due  at  settlement,  is  the  unpaid  balance  of  the  debt, 
$127.45,  together  with  the  interest,  $8.30. 

EXAMPLES. 

1.  Sold  goods  to  the  value  of  $650.39,  to  be  paid  Jan. 
27,  1848.     Required  the  amount  due  Aug.  19,  the  follow 
ing  payments  having  been  made  on  account :  Feb.  23,  $100; 
March  15,  $150.39  ;  May  20,  $200 ;  July  31,  $125. 

Ans.  $86.89. 

2.  A  Georgia  merchant  bought  goods  to  the  amount  of 
$575,  and  obtained  credit  till  Jan.  31,  1849.     But  being 
unable  to  pay  the  whole  debt  at  one  time,  he  remitted  8100 
when  it  became  due,  and  subsequently  paid  $75  March  3, 
$100  March  27,  $150  April  17,  and  the  balance  June  7. 
What  was  the  amount  of  the  last  payment,  the  legal  rate 
of  interest  being  8  per  cent.  ?  Ans.  $158.51. 

67.    LEGAL  INTEREST. 

The  rate  of  interest  varies  in  different  States  of  the 
Union.  In  the  examples  given  in  most  American  works 
on  Arithmetic,  6  per  cent,  is  understood,  unless  some  other 
rate  is  specified. 


220  THE   COUNTING-HOUSE.  [ART.  XV. 

In  each  of  the  New  England  States,  in  New  Jersey, 
Pennsylvania,  Delaware,  Maryland,  Virginia,  North  Caro 
lina,  Tennessee,  Kentucky,  Ohio,  Indiana,  Illinois,  Missouri, 
Arkansas,  and  the  District  of  Columbia,  and  on  U.  S.  notes, 
the  rate  is  6  per  cent.  In  New  York,  South  Carolina, 
Michigan,  Wisconsin,  and  Iowa,  it  is  7  per  cent.  In 
Georgia,  Alabama,  Mississippi,  Texas,  and  Florida,  8  per 
cent.  In  Louisiana,  5  per  cent.,  though  the  bank  interest 
is  .06,  and  conventional  interest  may  be  as  high  as  .10.  In 
Maryland,  the  interest  on  tobacco  contracts  is  .08.  In  Mis 
sissippi,  Missouri,  and  Arkansas,  the  interest  by  agreement 
may  be  as  high  as  .10,  and  in  Illinois,  Wisconsin,  and  Iowa, 
as  high  as  .12. 

In  the  mercantile  method  of  computing  interest,  com 
pound  interest  is  sometimes  charged.  But  as  the  courts  do 
not  generally  allow  compound  interest,  the  following  rule  is 
recommended  in  computing  interest  on  notes  and  bonds : 

If  any  payment  exceeds  the  interest  due  at  the  time  it  is 
made,  deduct  it  from  the  AMOUNT,*  and  compute  the  sub 
sequent  interest  on  the  balance.  If  the  payment  is  less  than 
the  interest,  deduct  it  from  the  INTEREST,  reserve  the  excess 
of  interest  to  be  added  to  the  succeeding  interest,  and  con 
tinue  the  interest  on  the  former  principal* 

a  The  amount  of  principal  and  interest. 

b  "  In  casting  interest  on  notes,  bonds,  &c.,  upon  which  partial 
payments  have  been  made,  every  payment  is  to  be  first  appropriated 
to  keep  down  the  interest ;  but  the  interest  is  never  allowed  to  form 
a  part  of  the  principal,  so  as  to  carry  interest.  27  Mass.  R.  417 ;  1  Dall. 
378. 

"  When  a  partial  payment  exceeds  the  amount  of  interest  due  when 
it  is  made,  it  is  correct  to  compute  the  interest  to  the  time  of  the  first 
payment,  add  it  to  the  principal,  subtract  the  payment,  cast  interest 
on  the  remainder  to  the  time  of  the  second  payment,  add  it  to  the 
remainder,  and  subtract  the  second  payment,  and  in  like  manner  from 
one  payment  to  another,  until  the  time  of  judgment.  1  Pick.  194; 
4  Hen.  &  Munf.  431 ;  8  Strg.  &  Rawle,  458  ;  2  Wash.  C.  C.  R.  167 ; 
see  3  Wash.  C.  C.  R.  350 ;  Ibid.  376. 

"  When  a  partial  payment  is  made  before  the  debt  is  due,  it  cannot 


§67.]  LEGAL   INTEREST.  221 


EXAMPLE  FOK  ILLUSTRATION. 


$1000.00  Philadelphia,  March  4th,  1841. 

For  value  received,  I  promise  to  pay  John  Smith,  or  order,  one 
thousand  dollars  on  demand,  without  defalcation.1 

WILLIAM  BROWN. 

December  1st,  1841,  received  $75.00.  July  17th,  1842,  re 
ceived  $15.50.  August  18th,  1843,  received  $30.50.  December 
llth,  1843,  received  $500.00.  January  3d,  1844,  received  $150.00. 
What  was  due  on  the  note,  Aug.  18,  1844? 

First  principal,  on  interest  from  March  4,  1841        .     $1000.00 
Interest  to  Dec.  1,  1841  (8mo.  27dy.)       .        .        .          44.50 

Amount        .         .     $1044.50 
First  payment,  exceeding  the  interest  due        .         .  75.00 

Balance  for  a  new  principal 969.50 

Interest  from  Dec.  1,  1841,  to  July  17,  1842 

(7mo.  16dy.) 36.52 

Second  payment,  less  than  interest  due          .  15.50 

Excess  of  interest  .         .         .         .  21.02 

Interest  from  July  17,  1842,  to  Aug.  18,  1843 

(13mo.  Idy.)         .....  63.18 

Interest  due  Aug.  18,  1843      .         .         .  84.20 

Third  payment,  less  than  interest  due  .  30.50 

Excess  of  interest  .         .         .         .  53.70 

Interest  from  Aug.  18,  1843,  to  Dec.  11,  1843 

(3mo.23dy.) 18.26       71.96 

Amount  due  Dec.  11,  1843       ....  1041.46 

Fourth  payment,  exceeding  the  interest  due  .  500.00 

Balance  for  a  new  principal  ,  541.46 

be  apportioned,  part  to  the  debt  and  part  to  the  interest.  As  if  there 
be  a  bond  for  one  hundred  dollars,  payable  in  one  year,  and  at  the 
expiration  of  six  months  fifty  dollars  be  paid  in.  This  payment  shall 
not  be  apportioned,  part  to  the  principal  and  part  to  the  interest,  but 
at  the  end  of  the  year  interest  shall  be  charged  on  the  whole  sum,  and 
the  obligor  shall  receive  credit  for  the  interest  of  fifty  dollars  for  six 
months.  1  Dall.  124." — Bouvier's  Law  Diet.,  vol.  1,  p.  717. 

*  The  laws  of  Pennsylvania  require  the  insertion  of  the  words 
"  without  defalcation."  In  other  places,  they  are  usually  omitted. 


222  THE   COUNTING-HOUSE.  [ART.  XV. 

New  Principal "541.46 

Interest  from  Dec.  11,  1843,  to  Jan.  3,  1844  (23dy.)  2.07 

Amount  due  Jan.  3,  1844 543.53 

Fifth  payment,  exceeding  the  interest  due  .  150.00 

Balance  for  a  new  principal  ....  393.53 

Interest  from  Jan.  3,  1844,  to  Aug.  18,  1844  (7mo.  15dy.)         14.76 

Amount  due  Aug.  18,  1844      .         .         .         .  $408.29 

EXAMPLES.1 

1.  Worcester,  July  4th,  1840. 
For  value  received,  I  promise  to  pay  Thomas  Jackson,  or 

order,  six  hundred  and  thirty-nine  dollars,  on  demand.5 
$639^0"  JoHN  WINTER- 

Endorsements.  Sept.  5,  1840,  received  $13.25.  Jan.  1,  1841, 
received  $1.50.  March  17,  1841,  received  $72.00.  Oct.  3,  1841, 
received  $29.50.  July  3,  1842,  received  $9.00.  What  was  due, 
Jan.  1,  1843?  Ans.  $601.83. 

2.  Portland,  May  13th,  1841. 
For  value  received,  I  promise  to  pay  George  Appleton, 

or  order,  nine  hundred  dollars,  on  demand. 

$900.00  WILLIAM  MASON. 

Endorsements.     Aug.   28,    1843,  received   $175.00.     Dec.   13, 

1843,  received  $10.00.    April  13,  1844,  received  $10.00.    May  1, 

1844,  received  $500.00.     What  was  due  Sept.  13,  1844? 

Ans.  $371.11. 

3.  New  York,  Oct.  5th,  1823. 
For  value  received,  I  promise  to  pay  Brown  &  Oliver,  or 

order,  one  thousand  dollars  on  demand. 

jiooo^o  JAMES  THOMAS- 


Endorsements.     March  4,  1824,  received   $100.00.     July   27, 
1825,  received  $50.00.     Oct.  25,  1825,  received  $100.00.     April 

*  For  the  rate  of  interest,  see  p.  220. 

b  In  some  of  the  States,  a  note  payable  on  demand,  does  not  draw 
interest  until  demand  is  made. 


§68.]  EQUATION    OP  PAYMENTS.  223 

13,  1826,  received  $15.00.  Nov.  13,  1826,  received  $10.00.  Dec. 
1,  1826.  received  $500.00.  What  was  due  on  the  note,  May  16, 
1830?  Am.  $532.70. 

4.  Charleston,  Aug.  18th,  1840. 

For  value  received,  I  promise  to  pay  Nathan  J.  Wilson, 
or  order,  four  hundred  and  thirty-one  dollars  in  six  months, 
with  interest  afterward.  EDWARD  ELLIS. 


$431.00 

Endorsements.  Feb.  18,  1841,  received  $31.00.  Sept.  15, 
1841,  received  $10.00.  Nov.  11,  1841,  received  $5.00.  March 
29,  1842,  received  $100.00.  May  13,  1843,  received  $200.00. 
Dec.  31,  1843,  received  $2.50.  What  was  due  June  16,  1844? 

Ans.  $149.30. 

5.  Cincinnati,  Nov.  1st,  1841. 

For  value  received,  we  promise  to  pay  Samuel  Jones,  or 
order,  seven  hundred  and  seventy -five  dollars  and  fifty  cents, 
in  two  months  from  date.  HENRY  THOMPSON  &  Co. 


$775.50 

Endorsements.     Feb.  27,  1842,  received  $15.00.     August  20, 

1842,  received  $15.00.     Dec.  18,  1842,  received  $15.00.     Jan.  1, 

1843,  received  $15.00.     April  1,  1843,  received  $200.00.     Feb. 
19,  1844,  received  $21.00.     What  was  the  balance  due  Oct.  1, 
1844?  Ans.  $603.64. 

68.    EQUATION  OF  PAYMENTS. 

The  ordinary  rule  for  equation  of  payments,  is  founded 
on  the  supposition  that  the  interest  of  the  money  which  is 
not  paid  until  after  it  is  due,  is  equal  to  the  discount  of 
that  which  is  paid  before  it  is  due.  This  is  not  strictly 
correct,  but  it  corresponds  to  the  usual  method  of  computing 
discount  among  merchants. 

The  labor  of  equating  may  be  abbreviated  by  disregarding 
the  cents  if  they  are  less  than  50,  and  counting  them  as  an 
additional  dollar  if  they  are  more  than  50.  When  the  sums 
are  all  large,  the  units  of  dollars  may  be  disregarded  in  a 
similar  manner. 


224  THE    COUNTING-HOUSE.  [ART.  XV, 

To  find  the  equated  time  for  the  payment  of  several  debts . 

Multiply  each  charge  by  the  time  which  has  elapsed  from 
the  date  of  the  first  bill,  and  divide  the  sum  of  the  products 
by  the  sum  of  the  bills. 

To  find  the  equated  time  for  the  settlement  of  an  account, 
in  which  there  are  both  debits  and  credits  : — 

1st.  Find  the  equated  time  for  each  side  of  the  account. 

2d.  Multiply  the  least  side  of  the  account  by  the  time 
between  the  dates  found  for  each  side,  and  divide  the  pro 
duct  by  the  balance  of  the  account.  Tlie  quotient  will  be 
the  time  between  the  date  found  for  the  larger  side  of  the 
account,  and  the  equated  time  for  the  settlement  of  the 
balance. 

If  the  date  found  for  the  larger  amount  is  the  earliest, 
count  back,  but  if  it  is  the  latest,  count  forward  from  that 
date. 

EXAMPLE  FOR  ILLUSTRATION. 

Required  the  time  when  the  balance  of  the  following  account 
became  due : 

Dr.          Martin  Acres  in  Account  with  Simon  Gray.      Cr. 


1849 
June  14 
Aug.  18 
Sept.  22 
Dec.  18 

To  Balance 
"    Mdse. 
"   Bill  due 
"         Do. 

$875 
519 
289 
318 

30 
40 
37 
00 

1849 
July  10 
Sept.  16 
Nov.  18 
Dec.  20 

By  Cash 
"    Mdse. 
"    Draft 

"    Cash 

$600 
427 
350 
275 

00 
90 

DO 
00 

Drs. 

875.30  x  0  =  0 
519.40  x  65  =  33761. 
289.37  X  100  =  28937. 
318.  X  187  =  59466. 


Crs. 

600.00  x     26  =  15600. 
427.90  X     94  =  40222.60 
350.      x  157  =  54950. 
275.       x  189  =  51975. 


2002.07  122164.  1652.90  162747.60 

122164.  -j-  2002.07  =  61.  162747.60  -4-  1652.90  =  98. 

Debits  due  61  days  after  June  14.  Credits  due  98  days  after  June  14. 

1652.90  x  37 -j-  349.17  =175.  The  balance  was  therefore  due 
175  days  before  the  date  found  for  the  Dr.  side,  or  114  days  before 
June  14,  which  was  Feb.  20. 


§68.] 


EQUATION    OF   PAYMENTS. 


225 


The  example  may  be  equated  as  follows,  by  disregarding  the 
cents  and  units  of  dollars.  If  the  dollars  amount  to  5  or  more,  1 
should  be  added  to  the  number  of  eagles. 

88  x      0  =        0  60  x    26  =  1560 

52  x     65  =  3380  43  x     94  =  4042 

29  x  100  =  2900  35  x  157  =  5495 

32  x  187  =  5984  28  x  189  =  5292 


201  12264 

12264-4-201=61. 


166 
16389-^-166 


16389 


98. 


1652.90  x  37 -T-  349.17  =  175,  as  before. 


EXAMPLES  FOR  THE  PUPIL. 

1.  When  was  the  balance  of  the  following  account  due? 
Dr.      James  Day  in  Account  with  William  Knight.       Cr. 


ifc>48 

1848 

Jan.     6 

To  Mdse. 

§427 

20 

Feb.  20 

By  Cash 

§750 

00 

Mar.  10 

"     Do. 

316 

19 

Apl.  1C 

"    Mdse. 

95 

87 

May  18 

"     Do. 

284 

72 

July     3 

"    Draft 

100 

00 

Ans.  Nov.  29,  1847. 

2.  Find  the  equated  Ihne  for  the  settlement  of  the  fol 
lowing  account : 

Dr.  North,  Harrison  &  Co.  in  Ace.  with  J.  E.   Oliver.   Cr. 


1846 

1840 

I 

Jan.      1 

To  Balance 

§283 

94 

Jan.   12 

By  Cash 

§425  00 

May  15 

"   Mdse. 

217 

33 

Apl.     3 

"    Mdse. 

693 

13 

July     5 

"   Note 

500 

00 

June  29 

u      n 

37 

59 

Aug.  30 

"    Mdse. 

894 

60 

Ans.  Dec.  21,  1846. 

The  average  price  of  a  mixture  consisting  of  several 
ingredients,  is  found  in  nearly  the  same  manner  as  the 
average  time  for  the  payment  of  several  debts.  The  method 
of  finding  the  average  price,  is  usually  called  MEDIAL  ALLI 
GATION.  For  Examples,  see  Sect.  74. 
15 


226 


THE  COUNTING-HOUSE. 


[ART.  XV. 


69.    ACCOUNTS  CURRENT. 

An  ACCOUNT  CURRENT  contains  a  statement  of  the  mer 
cantile  transactions  of  one  person  with  another.  On  the 
Dr.  side  are  placed  all  the  payments  made,  and  the  amounts 
of  merchandise  sold,  to  the  merchant  who  is  furnished  with 
the  account;  on  the  Cr.  side  are  entered  all  sums  received, 
and  the  amounts  of  merchandise  purchased  from  the  said 
merchant. 

To  facilitate  the  settlement  of  interest,  it  is  customary  to 
place  against  each  sum  the  number  of  days  that  elapse  from 
the  time  that  each  entry  becomes  due,  until  the  time  of 
rendering  the  account,  and  calculate  the  interest  on  each 
item.  The  balance  of  interest  is  entered  on  the  side  which 
has  the  greatest  amount  of  interest. 

EXAMPLE  TOR  ILLUSTRATION. 

Abbott    $    Clark,    Philadelphia,    in   Account    Current   with   Charles 
Ooodhue  $  Co.,  Boston. 

DR. 


1844 

Dol.  Cts. 

No.  days. 

Interest. 

June     1 
"      13 

«     25 
July    19 
Sept.     1 

To  balance  due  from  for 
mer  acct.    . 
To  amount  due  on  note  for 
goods 
To  merchandise 
To  90  bbls.  flour,  at  $4.37J 
To  balance  of  interest     . 

153.50 

400.00 
275.00 
393.75 

3.28 

92 

80 
68 
44 

2.354 

5.333 
3.116 

2.887 

13  690 

$1225.53 

10.412 

3.278 

1844 

CR. 

Dol.  Cts. 

No.  days. 

Interest. 

June  16 
"      30 
July   29 
Sept     1 

By  cash 
By  bill  of  Arnold  &  Brown 
By  cash 

375.00 
250.00 
525.00 

77 
63 
34 

4.812 
2.625 
2.975 

on  a  new  account 

75.53 

10.412 

$1225.53 

,69.] 


ACCOUNTS   CURRENT. 


227 


1.  Oliver  Marriott,  Savannah,  in  Account  Current 
Daniel  Clark,  Mobile. 

Dr. 


1844 

Dol.       ct». 

No.  of  days. 

Intereit 

Feb.  16 

To  balance  due  from  old 

account, 

91.75 

"   25 

To  merchandise, 

163.50 

Apl.  13 

To  merchandise, 

219.25 

May  16 

To  balance  of  interest, 

1844 

Cr. 

Dol.       cts. 

No.  of  days. 

Interest. 

Mar.  30 

By  cash, 

400.00 

May  16 

By  balance  to  new  acct. 

2.  Joseph  Mason  fy  Co.,  Saint  Louis,  in  Account  Cur 
rent  with  Thompson  fy  Brother,  Lexington. 
Dr. 


1844 

Jan.  13 
Feb.    3 
"    25 
Mar.  29 
Apl.     1 

To  sheeting, 
To  duck, 
To  cambric, 
To  sundries, 
To  balance  of  interest, 

Dol.        cts. 

131.50 

87.75 
240.00 
300.00 

No.  of  days. 

Interest. 

1844 

Jan.  29 
Mar.    3 
Mar.  18 
Apl.     1 

Cr. 

By  furs, 
By  bill, 
By  cash, 
By  balance  to  new  acct 

Dol.       cts. 

175.00 
200.00 

380.75 

No.  of  days. 

Interest. 

3.  Henry  Chatham,  Nashville,  in  Account  Current 

with  George  Hapgood  fy  Sons,  Baltimore. 

Dr. 


1844 

Dol.        cts. 

No.  of  days. 

Interest 

Apl.    5 

To  amount  due  on  note  for 

goods, 

500.00 

May  17 

To                 ditto, 

119.50 

"   28 

To  merchandise, 

87.25 

June  16 

To  sundries, 

63.00 

July    1 

To  balance  to  new  acct. 

1844 

Apl.     9 

Cr. 

By  cash, 

Dol.        cts. 

350.00 

No.  of  days. 

Interest. 

"    11 

By  tobacco, 

289.50 

"    23 

By  bill  on  Atkins  &  Jones, 

200.00 

July    1 

By  balance  of  interest, 

228 


THE   COUNTING-HOUSE. 


[ART.  XV. 


It  is  more  convenient  to  enter  the  product  of  each  entry 
by  the  number  of  days,  as  in  ordinary  Equation  of  Pay 
ments,  instead  of  computing  the  interest  on  each  amount. 
By  dividing  the  balance  of  products  by  6000,  we  at  once 
obtain  the  balance  of  interest  at  6  per  cent.  For  any  other 
rate  than  6  per  cent.,  the  result  may  be  increased  or 
diminished,  as  in  computing  Bank  Interest. 

EXAMPLE. 

Wilson,  Survilliers  $  Co.  in  Account  Current  with  James  N.  Martin. 
DR.  CR. 


1850 

$ 

et 

drs. 

1850 

$ 

f 

lys. 

Jan.  9 

I'aid  for  9001.  remitted 

June  1 

By  proceeds  of  sales  ren 

to  Brown.  Shipley  & 

dered  this  dav.    Value 

Co.,  at  1.09     ... 

43CO 

00 

143 

619120 

due  Nov.  6,  1850      . 

15141 

75 

157  2377294 

*  20 

Freight  on  30  cases  per 
St.  Nicholas    .     .     . 

17 

10 

131 

2227 

Feb.  6 

C.  H.,  charge  for  ware 
housing       .... 
Paid  duties,  and  C.  H. 

6 

60 

917 

storage  . 

327 

76 

ill 

373P2 

"    8 

"     Marine  Insurance 

ieo 

OS 

112 

17920 

1  per  cent.,  ^instead  of 

usual    charges,)     per 

agreement  .... 

151 

42 

5  per  ceut.  commission 

and      guarantee      on 
$15141.75  .... 

757 

09 

Numbers     from    credit 

side  

237721M 

3054870 

Amount     of     products, 
305-1  870  div.  by  60  - 

509 

IS 

Balance  dueW.  S.  &  Co. 

8852 

58 

15141 

--, 

"    " 

By  balance     .... 

8852 

->8 

As  the  amount  of  sales  in  the  preceding  account  is  not 
due,  Wilson,  Survilliers  &  Co.  should  be  charged  with 
interest  until  it  becomes  due ;  therefore  the  product  2377294 
is  added  to  the  amount  of  the  Dr.  products.  In  forming 
the  products,  the  cents  are  disregarded,  unless  they  exceed 
50,  in  which  case  the  number  of  dollars  is  increased  by  1. 

EXAMPLES. 

1.  A.  B.  in  account  with  C.  D.  Debits,  1849,  July  3, 
$5000;  July  28,  $40;  Aug.  16,  $800;  commission  and 
charges,  6°|0  on  $18000.  Credits,  1850,  March  2,  $9000 ; 
May  15,  $9000.  What  was  the  balance  of  account  rendered 
Jan.  1,  1850.  Ans.  $10617.89. 


§70.] 


PRACTICE. 


229 


2.  E.  F.  in  acct.  with  GL  H.  Debits,  1850,  Jan.  20, 
$3300;  Jan.  31,  $25;  Feb.  19,  $400;  June  8,  $900;  com 
mission  and  charges,  7°|0  on  $13842.68.  Credit,  proceeds 
of  sales,  due  Oct.  29,  $13842.68.  Required  the  balance 
due  July  1.  Ans.  $7869.86. 

The  pupil  may  solve  by  this  method  the  examples  given 
on  page  225. 

7O.    PRACTICE. 

In  PRACTICE  many  questions  arise  that  can  be  solved 
more  readily  than  by  adopting  any  of  the  ordinary  rules. 
Many  of  the  operations  of  business,  in  which  compound 
numbers  are  concerned,  may  be  abbreviated  by  first  finding 
values  for  the  highest  denomination,  and  considering  the 
lower  denominations  as  aliquot  parts  of  the  higher.1 


Of  a 

cts. 

50    -- 

333' 

25 

20 


dol. 


a 


10 
8] 
64 


=  A 


4    = 


2    = 


TABLE  OF  ALIQUOT  PARTS. 
Of  a 

qr.  Ib. 


Of  a£. 

s.  d.  £ 
10  =  ^ 
6  8=  | 
5  =  -] 
4  — •  1 
I 

6 


10= 


4  — 

* 


Of  a  ski. 
d.  s. 
6  =' 

4=3 

3  =  -]- 
^  i_ 

H=I 

-A 

3  1 

4  T5 

i^A 

i=A 

q    _3 

s=i 

«!  —  5 
7S—  8 


0/a^ 
c?y^.  g'r. 
10 

5       = 

4 

2     2  = 

2 

1     1  = 


ton. 


12 
16 
15 

7     2 


16 
14 


cwt. 

cwt. 


20=  I 


1 
2     8 

2  24 

3  12 


= 


Of  a  year 
m.  d.      y 

6  =  \ 
4      =i 
3      =  -j- 
212=  J- 
2      =  & 
115=  J 
1  10=  J 

424=fC 

7  6=  I 
918=   | 
415=§ 
715=  | 


a  When  the  number  of  articles  is  not  very  large,  and  the  price  con 
sists  of  several  denominations,  the  answer  can  generally  be  obtained 
most  readily  by  Compound  Multiplication. 

When  the  number  of  articles  is  large,  and  the  price  such  as  to 
require  but  few  parts  to  be  taken,  the  readiest  mode  of  solution  is 
by  Practice. 

When  the  quantity  and  price,  consist  each  of  several  denominations, 
the  best  mode  of  solution  is  by  Proportion,  or  by  Fractional  Analysis. 


280  THE   COUNTING-HOUSE.  [ART.  XV. 

Similar  tables  may  be  made  to  any  required  extent,  but 
these  are  sufficient  to  show  their  application. 

One  of  the  following  rules  may  be  adopted  in  nearly  all 
questions  that  admit  of  abbreviation  by  Practice  : 

1.  Assume  the  price  at  some  unit  higher  than  the  given 
price,  and  take  aliquot  parts  of  the  assumed  price  for  the 
answer. 

2.  Multiply  the  price  by  the  integers  of  the   highest 
denomination,  and  take  aliquot  parts  for  the  lower  denomi 
nations. 

In  the  application  of  either  of  these  rules,  there  is  great 
room  for  the  exercise  of  judgment,  in  determining  what  parts 
should  be  taken  to  determine  the  answer  most  readily. 

EXAMPLES  ILLUSTRATING  THE  FIRST  RULE. 

1.  At  43  f  cents  a  yard,  what  is  the  price  of  87  J  yards 
of  muslin  ? 

The  price  at  $1  per  yd.  would  be  $87.50 
at  25cts.  =  $J  21.875 

at!2J"  =  Jof  $J  10.9375 

at    6i  "  =  J  of  $i  5.46875 

at  43f  $38^28125"" 

Or,  as  $0.43f  =  $T7g,  the  answer  might  have  been  obtained 
by  multiplying  $87.50  by  T75. 

2.  What  is  the  value  of  96*  lb.   of  tea,  at  3s.   10|d. 
per  Ib.  ? 

The  price  at  £1  per  Ib.  would  be  £96  10s. 

at  3s.  4d.    =  J£ 
at        5d.    =  J  of  3s.  4d. 
at        lid.  =  i  of  5d. 
at  -d.  =  J    of  5d. 


at  3s.  10  jd. 


§70.]  PRACTICE.  231 

EXAMPLE  FOR  ILLUSTRATING  THE  SECOND  RULE. 
What  is  the  value  of  llcwt.  3qr.  17  Ib.  of  sugar,  at  £1 
3s.  6d.  per  cwt.  ? 

£     s.     d. 


2qr.  =  £cwt. 
16  Ib.  =  Jcwt. 

1  qr.  =  £  2  qr. 
lib.  =  Jg161b. 

1 

3 

6 
11 

price 

of 

Icwt. 

16  Ib. 
lib. 

12 

18 
11 
5 
3 

6 
9 
101 

2f 

price 
(t 

(i 

of 

ft 

u 

llcwt. 

2qr. 
Iqr. 

13 

19 

Si- 

1  price 

of 

llcwt.  3  qr. 

17  Ib. 

The  following  rule  is  probably  as  convenient  as  any  that 
could  be  given,  for  computing  interest  by  Practice. 

Multiply  1  per  cent,  of  the  principal  by  J  the  even  num 
ber  of  months,  and  if  there  is  an  odd  month,  add  30  to  the 
number  of  days.  Divide  the  days  by  6,  and  multiply  ^ 
of  1  per  cent  by  the  quotient.  If  there  are  any  remaining 
days,b  take  as  many  60ths  of  1  per  cent.  Add  the  numbers 
so  obtained,  and  their  sum  will  be  the  interest  at  6  per  cent. 
For  any  other  rate,  increase  or  diminish  the  result,  as  in 
the  Bank  Rule. 

EXAMPLE  FOR  ILLUSTRATION. 

What  is  the  interest  of  $9763.25  for  2yr.  9mo.  8dy.? 

1  per  cent.,  or  the  interest 
for    2    months,   ta    J97.6325.         j  0\ * 
Multiplying  by  16,  we  obtain 
the  interest  for  2yr.  8mo.    The  585.7950 

Q7fi  Q9^ 

Imo.  remaining,  added  to  the  ^  .g   Q  rf  ^        ™j^ 

8  days,  gives  38  days,  in  which  2dy>  ig    j    of  2mo>           3.2544 

6  is  contained  6  times,  with  2  

remainder.      As   the   interest  $1623.9539 

a  The  fraction  is  g^,  but  as  nothing  is  reckoned  less  than  $d,,  it  is 
called  £. 

b  When  the  remainder  is  4,  it  is  more  convenient  to  diminish  the 
quotient  by  1,  and  call  the  remainder  10,  taking  g$  or  ^  of  1  per  cent 
for  the  interest. 


232  THE    COUNTING-HOUSE.  [ART.  XV. 

for  6  days  is  .1  of  1  per  cent., or  $9.763-|-,  the  interest  for  36  days 
is  6  times  as  much.  The  two  remaining  days  are  -^  of  60  days, 
we  therefore  add  ^j  of  §97.634-. 

There  are  a  variety  of  other  contractions  that  may  frequently 
be  adopted  in  practice.  A  few  are  given  below,  which  will  often 
be  found  useful. 

(1.)  When  the  multiplier  consists  of  any  number  of  9's,  increase 
it  by  1,  and  subtract  the  multiplicand  from  the  product.  Thus, 
18473x9999=184730000—18473  =  184711527. 

(2.)  To  multiply  by  5,  divide  the  multiplicand  by  .2.  Thus, 
187  X  5  =  187  -f-  .2  =  935.  To  divide  by  5,  multiply  the  dividend 
by  .2. 

(3.)  To  multiply  by  25,  divide  the  multiplicand  by  .04.  Thus, 
1289  X  25  =  1289  -j-  .04  =  32225.  To  divide  by  25,  multiply  the 
dividend  by  .04. 

(4.)  To  multiply  by  75,  multiply  by  100,  and  subtract  £  of  the 
product.  Thus,  18645  x  75  =  1864500  —  466125=  1398375.  To 
divide  by  75,  divide  by  100,  and  add  J  of  the  quotient. 

(5.)  To  multiply  by  125,  divide  the  multiplicand  by  .008.  Thus, 
1641  X  125  =  1641  -f.  .008  =  205125.  To  divide  by  125,  multiply 
the  dividend  by  .008. 

(6.)  To  multiply  by  375,  divide  by  .008,  and  multiply  the  quo 
tient  by  3.  Thus,  294  x  375  =  294  -:-  .008  X  3  =  110250.  To 
divide  by  375,  multiply  by  .008  and  divide  by  3. 

(7.)  To  multiply  by  625,  divide  the  multiplicand  by  .0016. 
Thus,  4812  X  625  =  4812  -f-  .0016  =  3007500.  To  divide  by  625, 
multiply  the  dividend  by  .0016. 

(8.)  To  multiply  by  875,  multiply  by  1000,  and  subtract  -J  of  the 
product.  Thus,  735  X  875=  735000  —  91875=  643125.  To  di 
vide  by  875,  divide  by  1000  and  add  -^  of  the  quotient. 

(9.)  To  multiply  by  any  number  within  12  of  100,  1000,  &c., 
annex  to  the  multiplicand  as  many  zeros  as  there  are  figures  in 
the  multiplier,  and  subtract  as  many  times  the  multiplicand  as  are 
equivalent  to  the  excess  of  100,  1000,  &c.,  over  the  multiplier. 
Thus,  24796  X  99989  =  2479600000  —  (11  X  24796)  =  2479327244. 

(10.)  To  square  a  number  ending  in  5,  multiply  the  number  of 
tens  by  one  more  than  itself,  and  place  25  at  the  right  of  the 
product.  Thus,  3X4  =  12,  and  35  X  35  =  1225;  12X13  =  156, 
and  125X125  =  15625;  6X7  =  42,  and  65  X  65  =  4225. 


§  70.]  PRACTICE.  233 

(11.)  When  the  tens  in  two  numbers  are  alike,  and  the  sum  of 
the  units  is  10,  to  obtain  the  product  multiply  the  number  of  tens 
by  one  more  than  itself  for  the  hundreds,  and  place  the  product 
of  the  units  at  the  right  of  this  product,  for  the  tens  and  units. 
Thus,  4  X  5  =  20,  and  43  X  47  =  2021;  42  X  48  =  2016;  44  X  46 
=  2024 ;  7  X  8  =  56,  and  72  X  78  =  5616  ;  71  X  79  =  5609,  &c. 

(12.)  The  sum  of  two  numbers  multiplied  by  their  difference,  is 
equal  to  the  difference  of  their  squares.  Hence  we  may  readily 
find  the  product  of  two  numbers,  one  of  which  is  as  much  above 
as  the  other  is  below,  a  certain  number  of  tens.  Thus,  87  X  73 
=  (80  +  7)  X  (80  —  7)  =  802  —  72  =  6400  —  49  =  6351. 

(13.)  To  square  any  number  between  50  and  60,  add  the  units 
of  the  given  number  to  25  for  the  hundreds,  and  annex  the  square 
of  the  units  for  the  tens  and  units.  Thus,  for  the  square  of  51 ; 
25  +  1  =  26  hundreds,  and  1  X  1  =  1 ;  hence  51x51=  2601.  In 
like  manner  53  X  53  =  2809  ;  59  X  59  =  3481. 

(14.)  When  one  figure  of  the  multiplier  is  an  aliquot  part  of  one 
or  more  of  the  remaining  figures,  the  work  may  be  abbreviated 
as  in  the  following  example  : — 


Multiply  489. 

137  by  7261.8. 

We   see  at 

once  that  18 

489.137 

7261.8 

is  a  multiple 

of  6,  and  72 

2934822      =  prod,  by  6, 

is  a  multiple 
of  18.  There 

8804466  =prod.  by  3 
35217864        =  prod,  by  4 

X  6=  prod,  by  18. 
X  18  =  prod,  by  72. 

fore,     multi 

3552015.0666 

plying      first 

by  6,  we  take  3 

times  the  product  for  the  j 

>roduct  by  18,  and 

4 

times  the  product  by  18,  for  the  product  by  72. 

(15.)  In  the  ordinary  mode  of  determining  the  greatest  common 
divisor  of  two  numbers,  any  prime  factor  or  square  number,  con 
tained  in  one  number  but  not  in  the  other,  or  any  prime  factor  or 
square  number  in  a  remainder  that  is  not  in  the  preceding  divisor, 
may  be  rejected,  and  the  work  thus  abbreviated.  For  example, 
let  the  greatest  common  measure  of  689  and  2279  be  required. 


234  THE   COUNTING-HOUSE.  [ART.  XV. 

Here  the  square  number  4  is  a 

factor  of  212,  and  not  of  689.    We  689  )  2279  (  3 

therefore  divide  212  by  4,  and  im 
mediately  obtain  the  greatest  com-  4 )  212 
men  measure      In  the  application           G_  —   ^ 
of  this  principle  to  the  reduction  of  53 
fractions,  we  observe  that  53  divides 
689  13  times,  and  it  divides  212,  4 
times.    It  therefore  divides  3  X  689 
-f212  or  2279,  3  x  13  +  4  or  43 
times.     Therefore 


Reduce  -||-|  to  its  lowest  terms. 

457  )  563  ( 1 

457  Neither  2  nor  53  being  factors  of  457, 

the  fraction  is  already  in  its  lowest  terms. 
106  =  2  X  53 

EXAMPLES. 

1.  Find  the  cost  of  15 lb.  lOoz.  of  tea,  at  $.37 J  per  lb.; 
at  $.25;  at$.31t;  at$.43f;  at  $.50;  at  $.56i;  at  $.66|; 
at  $.87*. 

2.  What  is  the  price  of  89fyds.  of  broadcloth,  at  $4.75 
per  yd.?  Ans.  $426.31 1. 

3.  What  is  the  value  of  49 A.  3R.  15r.  of  land,  at  $125 
per  acre  ?  Ans.  $6230.47. 

4.  What  is  the  value  of  96yds.  3qr.  3na.  of  broadcloth, 
at  £1  2s.  6d.  per  yard?  Ans.  £109  Is.  ltd. 

5.  What  will  17T.  llcwt.  2qr.  21  lb.  of  iron  cost,  at 
$19.75  per  ton?  Ans.  $347.29. 

6.  What  is  the  cost  of  163A.  2R.  25r.  of  land,  at  $15.75 
per  acre?  Ans.  $2577.59. 

7.  What  is  the  value  of  364yds.  3qr.  Ina.  of  sheeting, 
at  12 Jets,  a  yard?  Ans.  $45.60. 

8.  Bought  76bu.  3pk.  of  potatoes,  at  37 Jets,  a  bushel; 
19bu.  2pk.  of  wheat,  at  $1.10  a  bushel;  37bu.  Ipk.  of 


§  70.]  PRACTICE.  235 

barley,  at  62$cts.  a  bushel;  and  10T.   15ewt,  of  hay,  at 
$16,00  a  ton.     What  was  the  amount  of  the  whole? 

AKS.  $245.51. 

9.  What  is  the  interest  of  $2375  for  5yr.  llmo.  23dy.,  at 
.055  per  year?  Avts.  $781.21. 

10.  What  is  the  interest  of  $4814.25  for  Syr.  7mo.  14dy., 
at  .06  per  year  ?     At  .075  per  year  ? 

An*.  $1046.297;  $1307.87. 

11.  Multiply  576.3  by  99;  by  999000. 

12.  Multiply  7.894  by  5;  by  25;  by  7500. 

13.  Multiply  48.302  by  1250;  by  375000. 

14.  Divide  1879.4  by  5;  by  250;  by  75. 

15.  Divide  4449.17  by  125;  by  375. 

16.  Multiply  3.0872  by  525125,  by  adding  three  partial 
products. 

17.  Multiply  41909  by  999625125,  in  the  most  expedi- 
tious  manner. 

18.  Multiply  89443  by  625  5  by  875. 

19.  Divide  141.982  by  625;  by  875. 

20.  Multiply  89443  by  625875. 

21.  Multiply  283172  by  9992;  by  991. 

22.  What  is  the  square  of  15  ?  of  85  ?  of  115  ? 

23.  What  is  the  product  of  73  X  T7?  12  X  18?  44  X 
46? 

24.  What  is  the  product  of  81  X  £9  ?  75  X  75  ?  34  x 
36? 

25.  What  is  the  product  of  16  X  24  ?  19  X  21  ?  35  x 
45?  89  x  71?  67  X  53?  78  x  82?  96  x  84?  113x107? 

112  x  128  ? 

26.  What  is  the  square  of  58?  of  56?  52?  55?  57? 
54? 


236  THE    COUNTING-HOUSE.  [ART.  XV 

27.  Find  the  greatest  common  divisor  of  804  and  938  j 
of  741  and  1083;  of  1343  and  1817. 

28.  Reduce  each  of  the  following  fractions  to  its  lowest 
terms:  |^-J,  y7*^,  fff,  ills?  iVrV 

29.  Multiply  476384  by  9995125625. 

30.  Required  the  interest  of  $1729.50  for  3yr.  7mo.  16dy., 
at  7  per  cent. 

7 1 .     CAUSE  AND  EFFECT. 

In  all  questions  which  can  be  solved  by  ratio  or  propor 
tion,  if  the  multiplying  terms  are  written  in  one  column, 
and  the  dividing  terms  in  another,  the  factors  common  to 
both  columns  may  be  cancelled,  and  the  answer  obtained  by 
dividing  the  product  of  the  remaining  factors  in  the  column 
of  multipliers,  by  the  product  of  the  remaining  factors  in  the 
column  of  divisors. 

The  terms  of  a  proportion  may  be  distinguished  into 
causes  and  effects,  and  the  alternate  products  of  either  cause 
by  the  other  effect,  are  equal.  For  example,  if  8  men  build 
6  rods  of  wall  in  a  day,  4  men  will  build  3  rods  in  the 
same  time.  Stating  the  proportion,  we  have 

men.      men.        rod.       rod. 

8  :  4  :  :  6  :  3 

cause,      cause.      effect,    effect. 

1 1 '       I 


The  product  of  the  extremes,  and  the  product  of  the 
means,  give  us  the  product  of  each  cause  by  the  effect  of 
the  other  cause.  If  then  we  write  each  effect  opposite  to 
its  cause,  our  multipliers  and  divisors  will  be  obtained  with 
out  difficulty. 

Men,  animals,  and  times,  are  evidently  causes,  because 
the  increase  of  either  of  them,  will  increase  the  effect 
produced. 


§71.]  CAUSE    AND    EFFECT.  237 

In  questions  of  freight,  we  may  regard  distances  and  bulk 
as  causes,  producing  money  for  their  effect. 

The  principal  of  a  sum  of  money  at  interest,  is  a  cause, 
and  the  interest  is  an  effect. 

A  little  practice  will  give  great  facility  in  distinguishing 
between  causes  and  effects,  in  all  cases  of  common  occur 
rence,  and  by  acquiring  readiness  in  making  this  distinction, 
a  vast  amount  of  labor  may  be  saved. 

In  arranging  the  terms  according  to  this  rule,  when  we 
reach  the  required  term,  or  term  of  demand,  its  place  may 
be  supplied  with  a  dash.  Then,  as  the  product  of  all  the 
factors  on  each  side  must  be  equal,  the  missing  term  may 
be  found  b}r  cancelling,  and  dividing  the  product  of  all  the 
numbers  on  the  side  opposite  to  the  dash,  by  the  product 
of  all  the  numbers  on  the  side  in  which  the  dash  is  included. 

EXAMPLES  FOR  ILLUSTRATION. 

1.  If  18  men,  in  6  days  of  8  hours,  build  a  wall  150  feet  long, 
2  feet  wide,  and  4  feet  high,  in  how  many  days  of  12  hours  will 
24  men  build  a  wall  200  feet  long,  3  feet  wide,  and  6  feet  high  ? 


Commencing  our  statements,  ^  men 
we  write  18  men,  6  days  of  8  w  days 
hours,  as  cause,  and  the  effect,  g  hours 


men. 
—  days. 
i£  hours 


long. 

wide, 
hih. 


which  is  a  wall  150ft.  long,  2ft.  lono- 

wide,  and  4  ft.  high,  on  the  op-         -*-> 
posite  side.     Opposite  to  each  of         & 

these  terms,  we  write  —  days,  12 

y  days  Aits. 
hours,  24  men,  as  cause,  and  200 

long,   3  wide,    6  high,  as  effect. 

Cancelling  the  like  factors,  we  have  but  3  X  3  on  the  side  of  the 
multipliers,  and  1  on  that  of  the  divisors.  3  X  3  -=-  1  is  therefore 
the  missing  term,  or  number  of  days  required.  If  either  term 
were  fractional,  the  denominator,  representing  a  divisor,  should 
be  transposed  to  the  opposite  side.  By  proceeding  in  this  man 
ner,  a  statement  may  be  made  as  soon  as  the  question  can  be 
proposed. 

2.  If  $27-J   buy  4 1  yards  of  cloth,  that  is  fyd.  wide,  how 


238 


THE   COUNTING-HOUSE. 


[ART.  XV. 


many  yards  of  like  quality,  that  is  f-yd.  wide,  may  be  bought  for 
$13  j  ? 


STATEMENT.  Reducing  the  mixed  numbers 

13-|  to  improper  fractions,  we  trans 

pose   the  denominators,  writing 
4|-  them   above    the  causes,  —  then 

|  cancel  and  divide  as  before. 

Ans.   -}-£  =  1  Avd. 


19 


EXAMPLES  FOR  THE  PUPIL. 

1.  How  many  men  in  18  months,  will  build  a  wall  that 
108  men  can  build  in  16  months  ? a 

N.  B.     The  effect  in  this  example  is  1  wall. 

2.  How  many  bushels  of  meal  will  serve  54  persons  12 
months,  if  15  persons  consume  12  bushels  in  2  months?5 

3.  If  27  men  build  a  cistern  30ft.  long,  16ft.  wide  and 
10ft.  high,  in  4  weeks,  by  working  5  days  in  a  week,  and  9 
hours  a  day,  how  many  men,  working  6  days  in  a  week,  and 
12  hours  a  day,  will  build  a  cistern  48ft.  long,  12ft.  wide, 
and  20ft.  high,  in  9  weeks  ? c 

4.  What  is  the  interest  of  $360  for  3y.  4mo.,  at  6  per 
cent.  ?d 

5.  If  $16  gain  $3  in  5  mov  how  much  ought  $24  to  gain 
in  lOmo.  ? e 

6.  If  the  freight  of  1800  Ib.  for  56  miles,  is  $1.50,  how 
far  may  IT.  4cwt.  12  Ib.  be  carried  for  $6.75  ?f 

7.  How   much   wheat,    at    $1.20    a   bushel,    must    be 


a.  _ 

18 

108 
16 

b54 
12 

15   C27 
2     4 

6 
12 
9 

d  1 
12 

360  e  16 
40    5 

24 
10 

f  1800 
56 

2700 

1 

12 

9 

.06   — 

3 

6.75 

1.50 

48 
12 
20 

30 
16 
10 

§71.]  CAUSE   AND   EFFECT.  239 

given  in  exchange  for  90  barrels  of  flour,  at  $4.75  per 
barrel?* 

8.  If  the  rent  of  19A.  3R.  of  land  is  £4  10s.,  what  will 
be  the  rent  of  73  £  A.  ?b 

9.  If  the  expenses  of  a  family  of  8  persons  are  $40  in 
10  weeks,  how  many  persons  can  be  supported  12  J  weeks 
for  $100  ?c 

10.  The  shadow  of  a  stick  that  is  5ft.  Gin.  high,  measures 
3ft.  4in.     What  is  the  height  of  a  tree  whose  shadow  mea 
sures  75ft.  at  the  same  time  ?  d 

11.  If  29£bu.  of  wheat  yield  1760bu.  in  5  years,  how 
much  will  45ibu.  yield  in  6  years  at  the  same  rate  ? 

12.  If  10  compositors,  in  2  days  of  10  hours,  set  66f 
pages  of  types,  each  page  containing  45  lines  of  50  letters, 
how  many  compositors  will  set  94  1  pages,  each  page  con 
taining  35  lines  of  40  letters,  in  2-f  days  of  8  hours  ? 

13.  If  19  men,  in  71|  days  of  10|  hours,  dig  a  trench 
41-Jyd.  long,  5f  ft.  deep,  and  7-Jft.  wide,  how  long  a  trench, 
that  is  8§ft  deep,  and  4T%ft.  wide,  will  11  men  dig  in  291  £ 
days  of  4|  hours?  Am.  50|||§f|yd. 

14.  If  18  men  in  9J  months  consume  flour  worth  $78.75, 
when  wheat  is  $1.12J  per  bushel,  how  many  months  will 
$145  supply  35  men  with  flour,  when  wheat  is  $1.00  per 
bushel?  Am. 


15.  If  2500  slates,  each  8  inches  long  and  5  inches  wide, 
will  cover  a  roof,  how  many  will  be  required  that  are  6  inches 
long  and  4  inches  wide  ?  Ans.  4166f  . 

16.  A  pile  of  wood  60ft.  long,  10ft.  high,  and  8ft.  thick, 


a  —        90 
1.20  4.75 

b    2     4 

c       2  | 

d    3|  2 
11  1  — 

79     147 

81- 

10  |  25 
100  I  40 


240  THE    COUNTING-HOUSE.  [ART.  XV. 

was  sold  for  8288  i.     What  would  be  the  price  of  a  pile 
£0ft.  long,  8ft.  high,  and  4ft.  thick,  at  the  same  rate? 

Ans.  $31|. 

17.  If  5  men,  by  working  8  hours  a  day  for  12  days,  can 
build  a  wall  40  rods  long,  2  feet  thick,  and  6  feet  high,  how 
many  men,  working  9  hours  a  day,  will  build  a  similar  wall, 
80  rods  long,  3  feet  thick,  and  8  feet  high,  in  4  days  ? 

Ans.  20  men. 

18.  If  7  men,  in  8  £  days,  by  working  9  hours  a  day,  can 
build  i  of  a  wall  that  is  to  be  raised  12  feet,  how  many  days 
must  11  men  work,  when  the  days  are  8  hours  long,  to  raise 
the  same  wall  5  feet?  Ans.  7|§|  days. 

19.  If  61000  gain  $11  -J  in  80  days,  how  much  will  82500 
gain  in  120  days,  at  the  same  rate  ? 

20.  If  a  man,  by  walking  3  miles  an  hour,  for  6  hours  a 
day,  can  accomplish  a  journey  in  12  days,  in  how  many 
days  would  a  man  walk  the  same  distance,  at  the  rate  of 
2z  miles  an  hour,  for  9  hours  a  day  ? 

21.  If  42  £  bushels  of  corn,  that  weighs  51 J  pounds  a 
bushel,  can  be  bought  with  23  bushels  of  wheat,  that  weighs 
56f  pounds  a  bushel,  how  much  corn,  weighing  60  pounds 
a   bushel,  would   be  equivalent   to  100  bushels  of  wheat 
that  weighs  54  pounds  a  bushel?  Ans.  ISO.y^jbu. 

22.  If  a  man  travels  240  miles  in  8  days,  when  the  days 
are  12  hours  long,  how  many  miles  will  he  travel  in  24 
days,  when  the  days  are  1C  hours  long  ? 

23.  If  the  freight  of  2T.  6cwt.  for  28  miles,  is  814.50, 
what  will  be  the  freight  of  9T.  4cwt.  for  96  miles  ? 

24.  If  4  men,  in  3  days  of  8  hours,  build  40  rods  of  wall, 
how  many  rods  will  18  men  build  in  5  days  of  9  hours  ? 

Ans.  3371  rods. 

25.  How  many  men,  in  24  days  of  16  hours,  will  do  three 
times  as  much  work  as  18  men  can  perform  in  32  days  of 
12  hours  ? 


§71.]  CAUSE   AN7D   EFFECT.  241 

26.  If  14  men,  in  5J  days,  by  working  8  hours  a  day, 
reap  38 £  acres  of  grain,  how  many  men  will  reap  37  J  acres' 
in  6f  days,  by  working  9  hours  a  day  ? 

27.  How  much  wheat,  that  weighs  60  Ib.    per  bushel, 
would   be  required  to  supply  a  garrison  of  1400  men  9 
months,  if  2800  bushels,  weighing  58  Ib.  per  bushel,  supply 
800  men  3 £  months?  Ans.  12180bu. 

28.  How  many  hours  a  day  must  15  men  work,  to  dig  a 
trench  400ft.  long,  6ft.  wide,  and  3ft.  deep,  in  187  J  days, 
if  72  men  can  dig  a  trench  250ft.  long,  8ft.  wide,  and  4ft. 
deep,  in  31  i  days,  by  working  7  hours  a  day? 

29.  How  many  men  can  be  furnished  with  4  suits  each, 
by  1140  yards  of  cloth  that  is  liyd.  wide,  if  2016  yards, 
iyd.  wide,  furnish  112  men  3  suits  apiece  ? 

30.  If  16  compositors  set  150  pages  of  types,  each  page 
containing  48  lines,  and  each  line  50  letters,  in  3  days  of  10 
hours,  how  many  compositors  will  be  required  to  set  500 
pages  of  72  lines  each,  and  45  letters  in  a  line,  in  6  days  of 
8  hours?  Ans.  45. 

31.  If  81700,  at  6  per  cent.,  yield  an  interest  of  $350 
in  a  given  time,  what  will  be  the  interest  of  $3900  at  7  per 
cent,  for  one  half  the  time  ? 

32.  If  30  reams  are  required  for  1500  pamphlets  of  10 
sheets  each,  how  many  reams  will   be  required  for   740 
pamphlets  of  12 £  sheets  each? 

33.  If  17  yards  of  serge,  that  is  f  wide,  are  required  to 
line  a  cloak  containing  9.75  yards  of  cloth  that  is  5J  quarters 
wide,  how  many  yards  of  a  yard  wide,  would  line  a  cloak 
containing  10.5  yards  of  cloth  6i  quarters  wide? 

Ans.  15§i|§yd. 

34.  If  a  cistern  discharges  83£  gallons  of  water  in  1.3 
hours,  how  much  will  it  discharge  in  6  3  hours? 

16 


242  THE   COUNTING-HOUSE.  [ART.  XV. 

35.  When  exchange  on  London  is  at  a  premium  of  9J 
per  cent.,  what  is  the  value  of  $1863.50,  in  English  money, 
the  par  of  exchange  being  £1  =  $4.44|  ? 

72.    EXCHANGE. 

The  term  EXCHANGE,  in  commerce,  is  generally  employed 
to  designate  that  species  of  mercantile  transactions,  by  which 
the  debts  of  individuals  residing  at  a  distance  from  their 
creditors,  are  cancelled  without  the  transmission  of  money. 

A  BILL  OF  EXCHANGE  is  an  order  addressed  to  some  per 
son  at  a  distance,  directing  him  to  pay  a  certain  sum  to  the 
person  in  whose  favor  the  bill  is  drawn,  or  to  his  order.  The 
person  who  draws  the  bill  is  called  the  drawer  ;  the  person 
in  whose  favor  it  is  drawn,  the  remitter  or  payee ;  the  per 
son  on  whom  it  is  drawn,  the  drawee.  The  drawee  is  also 
called  the  acceptor,  when  he  has  accepted,  or  engaged  to 
pay  the  bills. 

Though  bills  of  Exchange  are  originally  drawn  by  credit 
ors  on  their  debtors,  they  are  very  rarely  transmitted  directly, 
but  pass  from  hand  to  hand  like  any  other  circulating 
medium,  and  are  bought  and  sold  in  the  market.  When 
the  remitter  disposes  of  a  bill,  he  writes  his  name  on  the  back, 
and  is  termed  the  endorser.  If  he  endorses  in  favor  of  any 
particular  individual,  he  gives  a  special  endorsement,  and  such 
endorsee  must  also  endorse  the  bill  if  he  negotiates  it.  But 
if  the  endorsement  is  blank,  the  bill  may  be  passed  at  pleas 
ure  from  hand  to  hand.  Every  endorser,  as  well  as  the 
acceptor,  is  held  responsible  for  the  payment  of  the  bill,  and 
may  be  sued  for  its  recovery. 

INLAND,  or  DOMESTIC  EXCHANGE,  includes  the  commei 
cial  transactions  within  the  limits  of  one  country.  FOREIGN 
EXCHANGE  relates  to  the  transactions  of  one  country  witb 
another. 

The  TRUE  PAR  or  EXCHANGE  is  the  value  of  the  cur- 


§  72.]  EXCHANGE.  243 

rency  of  one  country  estimated  in  the  currency  of  another, 
by  comparing  the  quantity  of  gold  and  silver  in  their 
respective  coins.  The  exchange  with  England  apparently 
furnishes  an  exception  to  this  rule,  the  nominal  par  being 
$4.44|  per  <£,  while  the  actual  value  of  the  pound  sterling, 
which  is  the  real  par,  is  about  $4.87.  Hence,  exchange  on 
England  is  generally  said  to  be  from  8  to  10  per  cent,  above 
par. 

The  COURSE  OF  EXCHANGE,  or  the  fluctuation  above  or 
below  par,  depends  generally  on  the  amounts  due  between 
different  countries.  Thus,  when  the  debts  and  credits  be 
tween  two  countries  are  equal,  the  real  exchange  is  at  par. 
But  if  New  York  owes  London  more  than  London  owes 
New  York,  there  will  be  a  greater  demand  for  bills  on  Lon 
don,  and  this  demand  will  produce  a  rise  in  the  price,  or 
cause  the  bills  to  be  at  a  premium.  The  premium,  how 
ever,  can  never  exceed  the  cost  of  transporting  specie;  for 
if  it  did,  all  debts  would  be  paid  in  money  or  merchandise, 
instead  of  bills  of  exchange.  The  nominal  premium,  how 
ever,  may  exceed  the  cost  of  remitting  coin,  when  the  nomi 
nal  par  is  above  the  real  par. 

The  operation  of  Bills  of  Exchange,  may  be  explained 
by  a  single  example. 

If  A.  of  Boston  owes  B.  of  Paris,  and  C.  of  Paris  owes 
D.  of  Boston,  A.  purchases  in  the  market  a  ~bill  upon  Paris; 
that  is,  he  buys  of  D.  an  order  on  his  debtor  C.,  to  pay  A. 
or  his  order  the  amount  desired.  A.  endorses  the  bill,  and 
sends  it  to  B.,  who  receives  payment  from  C.  Thus  the  two 
debts  are  cancelled  by  a  single  remittance;  the  inconve 
nience  of  exporting  and  re-importing  coin  is  removed,  and 
all  danger  of  loss  is  obviated  by  sending  three  bills  (called 
the  First,  Second,  and  Third  of  Exchange),  either  of  which 
being  paid,  the  others  are  void. 

An  ACCEPTANCE  is  an  engagement  to  pay  the  amount  of 
the  bill,  and  may  be  either  absolute  or  qualified.  An  abso- 


244  THE   COUNTING-HOUSE.  [ART.  XV. 

lute  acceptance  binds  the  drawee  when  the  bill  becomes  due, 
and  in  making  it  the  drawee  usually  writes  "Accepted/' 
and  subscribes  his  name  at  the  bottom,  or  across  the  body 
of  the  bill.  A  qualified  acceptance  implies  some  condition, 
as  the  sale  of  merchandise,  &c.,  and  does  not  bind  the 
acceptor  until  the  condition  is  complied  with.  If  a  bill  is 
made  payable  at  a  certain  time  after  sight,  the  acceptance 
should  be  dated. 

A  bill  should  be  presented  for  payment  during  the  regu 
lar  hours  of  business,  on  the  day  it  becomes  due. 

When  acceptance  or  payment  has  been  refused,  the  holder 
should  give  immediate  notice  to  all  the  parties  whom  he 
intends  to  hold  responsible  for  the  payment  of  the  bill. 
This  notice  is  usually  accompanied  with  a  PROTEST,  which 
is  an  instrument  prepared  by  a  public  notary,  stating  that 
acceptance  or  payment  has  been  demanded  and  refused, 
and  that  the  holder  of  the  bill  intends  to  recover  any 
damages  which  he  may  sustain  in  consequence. 

In  some  places  on  the  continent  of  Europe,  banks  of 
deposit  are  established,  and  exchanges  are  frequently  made 
by  transferring  the  amounts  credited  on  the  books  of  the 
bank,  from  one  person  to  another.  The  deposits  on  which 
these  credits  are  based,  are  called  banco,  and  they  usually 
bear  a  premium  above  the  ordinary  currency  of  the  country. 
This  premium  is  called  the  agio. 

The  comparative  market  value  of  gold  and  silver  is  con 
stantly  varying,  and  the  mint  value  is  differently  estimated 
by  different  governments.  Thus,  in  England  the  relative 
worth  of  the  two  metals  is  as  1  to  14.29 ;  in  France  as  1  to 
15.52,  and  in  the  United  States  as  1  to  15.99.  In  Eng 
land,  silver  is  so  much  overvalued,  that  it  would  banish  the 
gold  coins  from  circulation,  were  there  not  a  statute  pro 
viding  that  on  ly  gold  shall  be  legal  tender  in  all  payments 
of  more  than  40  shillings.  The  relative  value  of  the  pre 
cious  metals  should  always  be  considered,  in  estimating  the 
true  par  of  exchange  with  any  country. 


§72.]  EXCHANGE.  245 

DOMESTIC  EXCHANGE. 

Inland  Exchange  is  usually  effected  by  checks  or  DRAFTS, 
similar  in  form  to  the  following : — 

$1275.25  Philadelphia,  June  3,  1850. 

Sixty  days  from  date,  pay  to  James  N.  Lewis, 
or  order,  Twelve  Hundred  and  Seventy-Five  Dollars  and 
Twenty-Five  Cents,  and  charge  the  same  to  my  account. 

WILLIAM  MORRIS. 
To  Markham  &  Jones, 

Merchants,  Cincinnati. 

The  premium  or  discount  on  drafts,  may  be  owing  either 
to  a  difference  in  the  value  of  the  circulating  medium,  or 
to  fluctuations  in  the  demand. 

The  English  denominations  of  shillings  and  pence,  are 
still  retained  in  this  country  to  some  extent.  At  the  for 
mation  of  the  Constitution,  the  continental  currency  had 
suffered  a  greater  depreciation  in  some  of  the  colonies  than 
in  others.  Thus,  while  a  pound  in  New  England  was  worth 
$3.33£,  in  Pennsylvania  it  was  but  $2.66f,  and  in  New 
York  but  $2.50.  The  value  in  Federal  Money,  of  the  old 
currencies  of  the  different  States,  is  as  follows: — 

A  shilling  of  New  England,  Virginia,  Kentucky,  or  Ten 
nessee,  is  16f  cents. 

A  shilling  of  New  York  or  North  Carolina,  is  12  J  cents. 

A  shilling  of  New  Jersey,  Pennsylvania,  Delaware,  or 
Maryland,  is  13  J  cents. 

A  shilling  of  South  Carolina  or  Georgia,  is  21$  cents. 

We  cannot  remind  teachers  too  often  of  the  signal  benefits  they 
may  confer  upon  their  pupils,  by  communicating  collateral  know- 


246  THE   COUNTING-HOUSE.  [ART.  XV. 

ledge  to  them  ;a  that  is,  such  knowledge  as  is  directly  connected 
with  the  subject  of  their  lessons,  though  rarely,  if  ever,  found  in 
a  text-book.  This  practice  should  be  commenced  with  a  child  the 
first  day  he  enters  the  school-room,  and  should  never  be  disconti 
nued  until  the  day  when,  for  the  last  time,  he  leaves  it.  If  teach 
ers  would  make  themselves  familiar  with  such  books  as  Miss 
Mayo's  Lessons  on  Objects  ;  Mrs.  Hamilton's  Questions  ;  the  first 
fifty  pages  of  Wilmsen's  Children's  Friend,  and  similar  works,  it 
would  be  impossible  for  them  to  keep  school,  or  even  to  hear  a 
recitation,  without  overflowing  with  information,  both  instructive 
and  delightful.  The  school-room  would  then  cease  to  be  a  place 
so  far  out  of  the  world ;  and  the  gulf  which  has  so  long  separated 
it  from  actual  life  would  be  bridged  over.  When  it  was  our  for 
tune  to  be  a  teacher  of  the  Greek  and  Latin  classics,  we  used  to 
think  it  as  much  a  part  of  our  daily  duty  to  be  prepared  with 
illustrative  anecdotes  and  historical  facts,  drawn  from  the  man 
ners  and  customs  of  other  nations  and  times,  in  order  to  render 
each  lesson  more  useful  and  interesting,  as  to  be  prepared  for 
translation  or  syntax.  The  whole  business  of  the  school-room, 
from  morning  till  night,  should,  in  this  way,  be  made  attractive 
and  profitable.  Children  do  love  information  which  is  adapted  to 
their  capacities,  and  they  will  desire  to  go  where  it  can  be  found, 
as  naturally  as  bees  to  flowers.  An  absurd  objection  is  sometimes 
urged  against  such  a  course ;  namely,  that  it  will  only  amuse 
children,  turn  what  should  be  toil  into  pastime,  and  create  a  dis 
relish  for  close,  pains-taking,  solitary  application.  This  objection 
is  theoretic  merely.  It  is  never  made  by  those  who  have  tried 
the  experiment.  It  is  urged  only  by  such  as  are  too  ignorant  or 
too  indolent  to  make  the  necessary  preparation.  Not  only  reason, 
but  experience,  proves  that  it  is  the  best  possible  means  of  kin 
dling  a  desire  for  knowledge  in  the  bosoms  of  the  young ;  and 
when  this  desire  is  once  kindled,  the  teacher  has  only  to  direct 
the  car,  instead  of  dragging  it. 

We  propose,  on  the  present  occasion,  to  give  a  specimen  of  the 
kind  of  instruction  we  mean ;  to  show,  by  an  example,  how  col 
lateral  knowledge  may  be  appropriately  introduced  to  illustrate 
and  enrich  the  matters  contained  in  the  text-book.  And  we  may 
remark,  in  passing,  that  it  is  strange  how  any  teacher  can  ever 
use  the  term  text-book,  without  being  reminded  that  it  is  only  a 
collection  of  texts,  which  it  is  his  duty  to  explain  and  illustrate. 

In  our  cities,   every  merchant  and  most  business  men  have 

a  The  remainder  of  this  Section  was  originally  published  in  the 
December  numbers  of  the  Common  School  Journal,  for  1847. 


§72.]  EXCHANGE.  247 

much  to  do  with  bills  of  exchange  and  promissory  notes.  In  the 
country,  too,  almost  every  man  has  something  to  do  with  notes  of 
hand,  either  as  promisor  or  payee,  endorser  or  endorsee.  If  a  man 
borrows  money,  he  makes  these  instruments ;  if  he  lends  money, 
he  receives  them.  Every  respectable  man  is  liable  to  be  on  a 
jury,  where  questions  respecting  this  class  of  securities  are  tried; 
and  no  man  is  so  poor,  so  ignorant,  or  so  far  outside  of  all  society, 
as  not  to  hear  conversations  about  them. 

Suppose,  then,  a  class  of  advanced  scholars,  whose  minds  have 
been  previously  awakened  by  a  proper  course  of  instruction,  to  be 
asked  in  what  way  they  suppose  that  commercial  transactions 
between  the  merchants  of  different  nations  are  carried  on.  The 
citizens  of  the  United  States,  for  instance,  send  abroad  their 
productions  to  different  quarters  of  the  world,  to  the  amount  of  a 
hundred  millions  of  dollars  or  more  annually.  In  what  manner 
do  they  receive  their  pay  ?  In  money,  or  otherwise  ?  Should  any 
one  say  in  money ;  then  explain  to  him  the  immense  trouble,  risk, 
and  expense,  of  bringing  a  hundred  millions  of  dollars  from  other 
countries,  across  the  ocean  to  this,  which  amount  must  soon  be 
sent  abroad  again  to  pay  for  foreign  productions  which  we  want. 

Here  the  historical  fact  may  be  stated,  that  we  learn  from  the 
Pandects,a  that,  when  a  Roman  capitalist  had  lent  money  to  a 
foreigner,  the  common  mode  of  collecting  the  debt  was,  to  send  a 
trusty  slave  to  the  foreign  country  to  receive  the  debt  and  its 
interest,  and  to  bring  them  home.  But  this  was  necessarily  both 
expensive  and  perilous. 

Some  may  suppose  that  money  may  not  be  remitted  for  the  set 
tlement  of  each  transaction,  but  that  the  traffic  may  be  carried 
on  by  barter.  One  merchant  may  send  flour,  and  receive  his  pay 
in  cutlery  ;  another  beef,  and  receive  broadcloth,  &c.  It  will  be 
easy  to  answer  any  suggestions  of  this  kind  by  showing,  that,  on 
such  a  plan,  each  man  would  have  to  trade  in  everything,  or,  at 
least,  in  a  great  variety  of  things,  and  with  a  great  number  of 
men.  But  the  same  man  could  never  trade  in  books,  leather,  jew 
elry,  iron-ware,  silks,  pork,  tea,  fish,  fruits,  logwood,  flour,  cotton, 
rice,  oil,  feathers,  coal,  hemp,  molasses,  indigo,  otter  skins,  &c. 
&c.  ;  or  if,  by  any  possibility,  one  man  could  trade  in  all  these 
things,  he  never  could  trade  with  all  parts  of  the  world  from  which 
they  come.  Barter,  therefore,  must  always  be  confined  to  a  small 
number  of  articles,  and  to  the  same  place. 

a  Pandects,  the  digest  of  the  civil  law  published  by  Justinian.  Ex 
plain  who  Justinian  was,  and  when  and  where  he  lived. 


248  THE    COUNTING-HOUSE.  [ART.  XV. 

Here  the  subject  may  be  dismissed,  for  the  first  day,  and  the 
children  sent  away  to  devise  or  to  ascertain  in  what  way  commercial 
transactions  between  different  nations  may  be  made  expeditious, 
safe,  and  cheap.  We  suggest  the  suspension  of  the  subject  at 
this  point,  because  we  deem  the  proper  course,  in  regard  to  all 
instruction,  to  be,  first,  to  awaken  the  child's  mind  to  a  sense  of 
the  necessity  or  desirableness  of  knowledge,  and  to  put  it  into  an 
inquiring  or  receptive  state ;  and  then,  secondly,  to  rectify  the 
views  which  his  unaided  judgment  may  suggest,  or  to  impart,  when 
necessary,  the  precise  knowledge  he  needs. 

At  subsequent  recitations,  let  the  subject  be  taken  up  again, 
and  either  the  pupils  or  the  teacher  will  explain  it  as  follows : — 

A.,  in  Boston,  is  about  to  ship  flour  for  the  Liverpool  market. 
B.,  in  Liverpool,  is  a  corna  merchant,  and  will  buy  A.'s  flour. 
At  the  time  of  this  transaction,  C.,  also  a  merchant  in  Boston, 
wants  cloths  from  D.,  a  manufacturer  in  Manchester.  When  A. 
ships  his  flour,  he  draws  a  bill  of  exchange  on  B.,  in  Liverpool, 
in  which  he  requests  B.  to  pay  to  himself,  or  to  some  other  person 
named  on  the  face  of  the  bill,  and  to  the  order  of  whoever  is  so 
named,  a  sum  stated,  supposed  to  be  the  value  of  the  flour  when 
it  shall  reach  Liverpool.  But  C.,  in  Boston,  who  wants  cloths  from 
D.,  in  Manchester,  has  no  money  in  England  to  pay  for  them;  C. 
therefore  buys  A.'s  bill  on  B.,  and  pays  for  it  in  our  money.  If 
the  bill  be  made  payable  to  A.'s  order,  A.  endorses  it  to  C. ;  and  C. 
then  endorses  it  to  D., — or  he  endorses  it  in  blank,  as  it  is  called, 
which  phrase  the  teacher  must  explain, — and  sends  it  to  his  agent 
or  correspondent  in  England.  When  the  bill  arrives  in  England, 
C.'s  agent  or  correspondent  presents  it  to  B.  ;  and,  the  flour 
having  arrived,  B.  accepts,  that  is,  promises  to  pay  it,  according 
to  its  terms.  It  is  then  taken  to  the  manufacturer  D.,  and,  the 
cloths  having  been  bought,  it  is  delivered  to  him.  D.  therefore 
becomes  B.'s  creditor,  and  receives  payment  from  him  in  English 
money,  as  A.  had  received  his  pay  from  C.  in  Boston  money.  Thus 
the  transaction  is  completed  without  the  trouble,  expense,  or  risk 
of  sending  a  cent  of  money  across  the  ocean,  to  be  sunk  by  storms, 
or  plundered  by  pirates. 

But  suppose,  in  the  above  case,  that  C.,  after  having  bought 

a  Let  it  be  explained  that  the  word  "  corn,"  in  England,  never  has 
the  same  signification  as  with  us.  Here,  it  is  commonly  used  for 
Indian  corn  or  maize  ;  but  in  England,  it  is  a  generic  term,  and  meana 
wheat,  barley,  and  other  cereal  grains. 


§72.]  EXCHANGE.  249 

A.'s  bill  of  exchange,  does  not  wish  to  use  it  for  three  months ; 
but  his  neighbor  E.  wants  it,  or  one  like  it,  to  be  used  immedi 
ately.  Is  there  any  way  by  which  C.  can  transfer  this  bill  to  E., 
receive  his  money,  and  so  have  the  use  of  it  for  the  three  months, — 
or  must  he  go  back  to  A.,  have  the  bargain  rescinded,  the  bill  can 
celled,  and  a  new  one  drawn  in  favor  of  E.  ?  There  is  such  a  way. 
When  a  bill  or  note  is  drawn  payable  to  the  order  of  any  one,  it  is 
payable  to  whomsoever  that  one  shall  order  it  to  be  paid.  In  the 
case  supposed,  therefore,  C.  has  only  to  write  his  name  on  the  bill, 
with  the  words,  "Pay  to  E.,"  and  E.  receives  C.'s  whole  interest 
in  it.  If  he  says,  "Pay  to  E.,  or  his  order,"  then  E.  may  order 
it  to  be  paid  to  any  one  else  ;  and  so  on.  It  is  this  transferability, 
or  quality  of  being  transferable  from  hand  to  hand,  that  makes 
bills  of  exchange  and  promissory  notes  negotiable  instruments. 
This  word  negotiable  is  an  important  one,  and  the  meaning  of  it 
should  be  precisely  understood. 

By  the  civil  law  of  the  European  continent,  bills  of  exchange 
and  promissory  notes  were  early  recognised  as  mercantile  instru 
ments,  and,  from  their  nature,  negotiable.  But  in  England  there 
was  a  strong  prejudice  against  the  assignment  or  transfer  of  debts, 
because  of  the  abuses  liable  to  be  practised,  if  one  man  could  buy 
up  a  debt  against  another,  and  sue  and  imprison  him  on  it.  There 
are  still  laws,  both  in  England  and  in  this  country,  against  buying 
up  debts  for  purposes  of  oppression.  Anciently,  the  common  law 
of  England  forbade  the  assignment  or  negotiation  of  promissory 
notes.  But  the  statutes  of  3  and  4  Anne  gave  negotiability  to 
notes,  placing  them,  as  mercantile  contracts,  on  the  same  footing 
us  inland  bills  of  exchange.  These  English  statutes  have  been 
generally  adopted  in  the  United  States,  as  a  part  of  our  common 
law. 

At  the  present  time,  therefore,  bills  of  exchange  and  promissory 
notes,  by  their  quality  of  negotiability,  are  the  means  by  which 
debts  and  credits  are  transferred  from  one  person  to  another,  with 
safety,  despatch,  and  economy.  They  afford  means  of  circulation 
for  all  the  property  they  represent,  and  thus  they  enlarge  in  every 
country  its  stock  of  circulating  wealth,  or  its  means  of  trade. 

Promissory  notes  are  of  two  kinds ; — negotiable,  and  not  negotiable. 

A  negotiable  note  expresses  on  its  face  that  it  is  payable,  not 
only  to  the  person  named  in  it,  but  to  any  other  person  who  shall 
acquire  the  legal  interest  in  it.  If  it  be  made  payable  to  John 
Stiles  or  order,  it  is  then  negotiable  by  endorsement ;  if  to  John 
Stiles  or  bearer,  it  is  then  negotiable  by  delivery. 


250  THE    COUNTING-HOUSE.  [ART.  XV. 

A  note  not  negotiable,  expresses  on  its  face  that  it  is  payable  to 
the  particular  person  named  in  it, — as  to  John  Stiles.  Such  a 
note  is  payable  only  to  the  party  named. 

All  valid  promissory  notes  import  a  valuable  consideration  ;  that 
is,  an  action  at  law  may  be  sustained  upon  them  without  specially 
setting  forth  or  proving  a  consideration  for  the  note.  In  this  they 
differ  from  other  unsealed  contracts. 

In  a  promissory  note,  there  are  two  original  parties, — the  maker 
of  the  note,  who  is  called  the  promisor ;  and  the  party  to  whom 
it  is  made  payable,  who  is  called  the  promisee  QY  payee. 

A  valid  negotiable  promissory  note  is  a  written  promise  for  the 
payment  of  money,  at  all  events. 

The  promise  must  be  in  writing,  but  it  may  be  in  ink  or  pencil ; 
and  all  but  the  signature  may  be  in  printed  letters.  The  signa 
ture  gives  efficacy  to  the  note,  and  must  be  in  the  handwriting  of 
the  promisor  or  of  his  authorized  agent. 

The  form  of  words  used  is  not  material,  provided  the  note 
contains  a  written  promise  to  pay.  A  mere  acknowledgment  of 
indebtedness  is  not  sufficient;  Thus,  "I  owe  you  $300,"  though 
in  writing,  is  only  a  due  bill ;  it  is  not  a  promissory  note. 

The  note  must  be  for  the  payment  of  money.  Therefore,  a 
written  promise  to  pay  in  goods  or  labor,  is  not  a  negotiable 
promissory  note,  although  put  into  the  form  of  a  note,  and 
payable  "to  order"  or  "bearer." 

A  negotiable  note  must,  on  its  face,  fix  and  make  certain  the 
amount  of  money  to  be  paid,  either  in  words  or  figures.  Hence, 
a  written  promise  to  pay  "  all  that  shall  be  due  on  final  settle 
ment,"  or  "  all  that  shall  be  realized  from  the  growing  crop,"  or 
"  all  that  shall  be  received  from  John  Stiles,"  is  not  a  negotiable 
promissory  note.  Even  though  a  part  of  the  sum  to  be  paid 
should  be  made  certain,  on  the  face  of  the  note,  it  is  yet  not  a 
negotiable  promissory  note,  even  for  the  part  which  is  so  made 
certain. 

The  money,  by  the  note  itself,  must  be  made  payable  at  all 
events,  and  independently  of  any  contingency.  Therefore,  a  written 
promise  to  pay  "when  certain  goods  are  sold,"  or  "when  a 
certain  ship  arrives,"  is  not  a  negotiable  promissory  note.  So,  if 
the  note  is  made  payable  out  of  a  particular  fund,  as  "my  next 
month's  wages,"  it  is  not  a  negotiable  promissory  note. 

And  the  promise  must  be  to  pay  money  on  a  day  certain, — a 

y  fixed  by  the  note  itself.     Therefore,  a  promise  to  pay  $100, 


§72.]  EXCHANGE.  251 

"•when  A  shall  come  of  age,"  is  not  a  negotiable  promissory 
note ;  for  A.  may  never  come  of  age.  But  a  note  promising  to 
pay  $100  when  A.  shall  die,  is  valid ;  for  A.'s  death  on  some  day 
is  certain ;  and  the  note,  by  its  own  terms,  fixes  that  day  for 
payment. 

A  note  must  contain  no  uncertainty  as  to  the  person  to  whom  it 
is  payable.  Hence,  a  written  promise  to  pay  to  "A.  or  B.,"  is 
not  a  negotiable  promissory  note.  But  a  written  promise  to  pay 
to  "A.  or  bearer,"  is  good;  for,  in  legal  effect,  such  a  note  is 
payable  to  "bearer;"  and  any  person  who  has  legal  possession  of 
the  note,  and  presents  it  for  payment,  is  the  "bearer"  intended. 

A  note  may  be  issued  with  a  blank  for  the  payee's  name  ;  and, 
in  such  case,  any  bond  fide  holder  may  fill  up  the  blank  with  his 
own  or  any  other  name,  and  the  note  will  then  be  treated  as 
though  it  had  been  valid  in  all  respects  from  its  date. 

It  is  indispensable  that  the  maker's  name  should  appear  on  the 
note  as  promisor.  The  name,  however,  may  be  written  in  ink  or 
pencil,  and  at  the  top,  or  bottom,  or  in  the  margin  of  the  paper. 

It  is  not  indispensable  to  the  validity  of  a  note  that  it  should 
be  dated,  because  it  is  allowable  to  show  the  time  when  it  was 
made,  by  evidence  extrinsic  to  the  note  itself;  but  this  is  always 
expensive,  often  difficult,  and  sometimes  impossible.  If  a  note 
be  postdated  or  antedated,  the  time  of  its  actual  issue  may 
always  be  shown,  when  required  for  substantial  justice. 

It  is  not  necessary  that  a  note  should  specify  any  place  of  pay 
ment  ;  but  when  it  is  the  intent  of  the  parties  that  it  shall  be 
paid  at  a  particular  place,  the  place  must  be  specified  in  the  body 
of  the  note.  A  memorandum  at  the  bottom  of  the  note,  or  on  its 
margin,  is  not  sufficient. 

Nor  is  it  essential  that  a  note  should  be  attested.  An  attesta 
tion,  however,  in  Massachusetts,  takes  a  note  out  of  the  statute 
of  limitations,  as  to  the  payee,  his  executor  or  administrator. 

A  promissory  note  may  be  made  by  one  person,  or  by  two  or 
more  persons.  When  made  by  two  or  more  persons,  it  may  be 
joint,  or  joint  and  several. 

When  two  or  more  persons  sign  a  note  written  thus:  "  We  pro 
mise  to  pay,"  &c.,  it  is  a  joint  note  only.  If  they  sign  a  note 
written  thus:  "I  promise  to  pay,"  &c.,  it  is  a  joint  and  several 
note.  When  a  note  is  joint,  all  the  promisors  must  be  jointly 
sued  ;  if  joint  and  several,  either  promisor  may  be  sued  alone. 

When  a  note  written  thus,  "We  promise,"  &c.,  is  signed  thus, 


252  THE   COUNTING-HOUSE.  [ART.  XV 

"A.  B.  as  principal,  and  C.  D.  as  surety,"  it  is  still  ihe  joint  note 
of  A.  B.  and  C.  D.  Had  it  been  written,  "I  promise,"  &c.,  and 
signed  in  the  same  way,  it  would  be  a  joint  and  several  note. 
The  words  "principal"  and  "surety"  only  show  the  relation  of 
the  makers  to  each  other ;  they  do  not  affect  other  parties. 

By  the  phrase  "negotiable  promissory  note,"  is  meant  an 
instrument,  negotiable  and  possessing  all  the  privileges  of  a 
promissory  note  in  commerce.  A  note  not  negotiable  is  never 
theless  a  binding  contract  between  the  parties  to  it. 

It  is  in  reference  to  the  transfer  of  a  note  from  hand  to  hand, — 
like  a  bank  bill  or  a  Bank  of  England  note, — that  the  question  of 
its  negotiability  becomes  material. 

THE  TRANSFER  OF  NOTES. 

A  note  may  be  transferred  by  delivery,  or  by  endorsement. 

As  TO  TRANSFER  BY  DELIVERY. — The  rule  is,  that  no  person 
whose  name  is  not  on  the  note,  as  a  party  thereto,  is  liable  on 
the  note. 

Therefore,  when  a  note  payable  to  bearer,  or  endorsed  in  blank, 
is  transferred  by  the  holder,  by  delivery  only,  the  party  transferring 
it  is  not  liable  upon  it. 

By  not  endorsing  it,  he  is  understood  to  mean  that  he  will  not 
be  responsible  on  it ;  and  such,  therefore,  is  the  contract  between 
him  and  the  party  receiving  it. 

But  if,  in  such  case,  the  note  is  received,  by  the  party  to  whom 
it  is  delivered,  as  a  conditional  payment  of  a  debt  previously  due 
him,  or  as  a  conditional  satisfaction  of  any  other  valuable  con 
sideration  then  given,  the  party  transferring  it,  if  the  note  is 
dishonored,  (that  is,  if  not  paid,)  on  legal  presentment  and  notice, 
will  be  responsible  for  the  debt,  or  consideration,  though  not 
directly  suable  on  the  note. 

And  though  a  party  transferring  a  note  by  delivery  only,  is  not 
liable  on  the  note,  he  is  not  exempt  from  all  obligations  or  respon 
sibilities. 

In  the  first  place,  by  legal  implication,  he  warrants  his  own 
title  to  the  note,  and  his  right  to  transfer  it  by  delivery. 

Then  he  warrants  that  the  note  is  genuine,  and  not  forged  or 
fictitious. 

And  he  warrants,  moreover,  that  he  has  no  knowledge  of  any 
facts  which  make  the  note  worthless ;  for  instance,  if  the  note 
be  a  bank  note,  and  the  party  transferring  it  knows  the  bank  has 
failed,  and  conceals  this  knowledge,  his  act  is  a  fraud,  and  the 


§72.]  EXCHANGE.  253 

consideration  he  received  may  be  recovered  back.  The  fraud 
makes  void  the  contract.  And  even  if  the  failure  of  the  bank, 
at  the  time  of  the  transfer,  was  unknown  to  either  of  the  parties 
to  it,  it  is  the  better  opinion  that  the  transferrer  must  bear  the 
loss,  because  it  is  implied  in  the  transaction  that  the  note  would 
be  paid  on  due  presentment. 

As  TO  TRANSFER  BY  ENDORSEMENT. — When  a  note  is  payable  to 
a  person,  or  his  order,  it  is  properly  transferable  only  by  endorse 
ment,  as  nothing  else  will  give  to  the  holder  a  legal  title,  so  that 
he  can,  at  law,  hold  the  parties  to  the  note  directly  liable  to  him. 

13y  a  mere  assignment  of  a  negotiable  note,  the  holder  acquires 
only  the  same  rights  that  the  assignment  would  give  him,  if  the 
note  were  not  negotiable. 

No  particular  form  of  words  is  required  to  make  an  endorsement 
legal ;  generally  it  is  enough  if  the  signature  of  the  endorser  is 
on  the  note,  without  any  words  at  all ;  and  this  is  the  usual  mode 
of  endorsing  notes. 

The  endorsement  may  be  on  either  side,  or  any  part  of  the  note, 
or  on  a  paper  annexed  to  it,  and  in  ink  or  in  pencil. 

A  note  transferable  by  delivery  only,  may  be  endorsed  ;  and 
then  the  endorser  incurs  the  same  obligations  and  liabilities  as  if 
the  note  had  been  originally  made  transferable  by  endorsement  only. 

The  time  of  endorsing  a  note  may  be  material,  for  if  a  person, 
(not  the  payee  of  a  negotiable  note,)  endorses  it  when  it  is  made,  he 
will  be  liable  at  all  events,  not  as  endorser,  but  as  guarantor. 
If  lie  endorses  it  afterwards,  (not  being  a  regular  endorser,)  he 
will  be  liable  if  his  act  is  founded  on  any  legal  consideration,  but 
not  otherwise. 

Every  endorser,  by  his  endorsement,  contracts  with  every  sub 
sequent  holder  of  the  note, — 

1.  That  the  instrument  itself,  and  the  signatures  antecedent  to 
his,  are  genuine.  2.  That  he,  (the  endorser,)  has  a  good  title  to 
the  note.  3.  That  he  is  competent  to  bind  himself  as  endorser  by 
his  endorsement.  4.  That  the  maker  is  competent  to  bind  himself 
as  maker,  and  will,  on  presentment,  pay  the  note.  5.  That  if,  when 
duly  presented,  it  is  not  paid  by  the  maker,  the  endorser,  on  due 
notice,  will  pay  it. 

An  endorsement  may  be  in  "  blank,"  or  "  in  full,"  or  "restrict 
ive,"  or  "general,"  or  "qualified,"  or  "conditional." 

A  "  blank"  endorsement  is  merely  the  name  of  the  endorser 
•written  on  the  note. 


254  THE   COUNTING-HOUSE.  [ART.  XV. 

After  such  an  endorsement,  a  note  may  be  transferred  by  delivery 
only,  and  be  circulated  like  a  bank  note ;  and  any  holder  may 
write  out,  over  the  endorser's  name,  the  contract  implied  by  law 
on  the  part  of  the  endorser,  and  sue  upon  it. 

An  endorsement  is  said  to  be  "in  full,"  when  it  mentions  the 
name  of  the  person  in  whose  favor  it  is  made,  and  then  the  en 
dorsee  can  transfer  his  interest  in  it  only  by  writing  his  own 
endorsement  on  it. 

In  order  to  make  an  endorsement  "restrictive,"  there  must  be 
express  words,  showing  that  intent;  as,  "  Pay  to  John  Stiles  only." 

An  endorsement  is  said  to  be  "  general,"  when  it  is  in  blank,  or 
payable  to  the  endorsee  or  order. 

A  "  qualified"  endorsement  is  one  which  affects  the  liability  of 
the  endorser,  but  not  the  negotiability  of  the  note ;  as  when  to 
the  endorsement  is  added,  "  without  recourse,"  or  "  at  endorsee's 
own  risk,"  &c. 

A  "  conditional"  endorsement  limits  the  validity  of  the  endorse 
ment  to  some  future  event,  and  may  be  either  precedent  or  sub 
sequent  ;  as,  1st,  "  Pay  John  Stiles  the  within  on  my  marriage  ;" 
or,  2d,  "  Pay  John  Stiles,  or  order,  the  within  in  six  months, 
unless  he  sooner  receives  it  from  my  agent." 

Whoever  receives  an  endorsed  note,  contracts  with  the  endorser, 
(and  if  there  are  many,  with  each  of  them,)  that  the  note  shall 
be  presented  to  the  promisor  for  payment  at  the  proper  time  ;  that 
no  extra  time  for  payment  shall  be  allowed ;  and  that  notice  of 
non-payment  shall  be  immediately  given  to  the  endorser  ;  and  a 
default  in  any  of  these  particulars  discharges  the  endorser. 

Due  presentment  for  payment  requires  that  the  note  should  be 
presented  as  soon  as  it  becomes  due.  If  the  holder  could  delay  a 
day,  he  might  two  days,  or  a  year ;  but  any  delay  may  injuriously 
affect  the  endorser,  and  his  remedy  against  other  persons.  There 
fore,  if  the  holder  of  the  note  does  not  present  it  to  the  promisor 
on  the  day  it  becomes  due,  the  endorsers  are  discharged. 

And  the  rule  is  so,  although  the  holder  received  the  note  so 
near  the  time  of  its  maturity  as  to  make  the  demand  in  legal  time 
impossible. 

Ajid  such  demand  for  payment  is  required  though  it  is  known 
that  the  maker  is  dead  or  an  insolvent. 

Where  a  note  is  made  payable  on  demand,  the  time  at  which 
payment  must  be  demanded,  depends  on  the  circumstances  of 
the  case,  the  rule  being  that  payment  must  be  demanded  in  rea 
sonable  time. 


§73.]  ARBITRATION   OP  EXCHANGE.  255 

And  in  Massachusetts,  by  statute,  the  endorser  is  excused,  if 
the  demand  for  payment  on  the  maker  is  not  made  within  sixty 
days  from  the  date  of  the  note. 

If  a  note  is  payable  generally,  that  is,  without  any  place  being 
designated,  it  may  be  presented  at  the  maker's  counting-house  or 
dwelling-house.  If  it  is  presented  at  the  counting-house,  it  must 
be  within  the  hours  in  which,  by  the  usage  of  the  city  or  place, 
counting-houses  are  kept  open ;  if  at  the  dwelling-house,  then  at 
hours  while  the  family  are  up,  and  the  maker  may  be  presumed 
not  to  have  gone  to  bed. 

And  where  a  note  is  made  payable  at  a  particular  place,  the 
demand  must  be  made  at  the  place  fixed,  as  well  as  at  the  proper 
time  ;  otherwise  the  endorser  is  discharged. 

Where  a  note  is  payable  by  a  partnership,  presentment  to  either 
of  the  partners  is  sufficient.  Where  the  promisors  are  only  joint 
contractors,  and  not  partners,  demand  must  be  made  on  each. 

The  demand  must  be  made  with  the  note  ;  and  if  any  particular 
bank  or  place  is  fixed  for  payment,  the  note  must  be  there,  in 
order  to  make  the  demand  valid. 

On  the  failure  of  the  maker  to  pay,  the  holder  must  give  due 
notice  of  it  to  each  party  liable  to  him ;  and  if  he  fails  to  do  so 
to  any  party,  such  party  is  discharged. 

And  when  the  endorser  live's  in  the  same  place  with  the  holder, 
notice  may  be  given  on  the  day  when  the  demand  was  made,  or 
the  day  after,  but  not  later. 

When  the  endorser  and  holder  live  in  different  towns,  the  notice 
may  be  by  mail,  by  special  messenger,  or  by  private  hand. 

And  notice  by  the  mail  on  the  day,  or  the  day  after,  is  good, 
but  not  later. 

Where  there  are  numerous  endorsers,  each  is  entitled  to  notice, 
and  each  is  to  give  notice  to  all  parties  prior  to  himself;  and 
each  endorser  has  the  next  day  after  receiving  notice,  in  which  to 
give  notice  to  any  prior  party  whom  he  seeks  to  hold  liable  to 
himself. 

73.    ARBITRATION  OF  EXCHANGE. 

Merchants  often  find  an  advantage  in  remitting  bills  cir- 
cuitously,  rather  than  directly  to  the  place  where  they  are 
due.  The  determination  of  the  value  of  such  remittances 


256  THE   COUNTING-HOUSE.  [ART.  XV. 

is  called  ARBITRATION  OF  EXCHANGE,  and  is  best  deter 
mined  by  the  CHAIN  RULE. 


EXAMPLE  FOR  ILLUSTRATION. 

A  French  merchant  wishes  to  pay  in  London  a  bill  of 
£1500.  How  many  francs  must  he  pay  to  procure  remit 
tances  through  Russia,  Hamburg,  and  Spain,  allowing 
£13  =  75  roubles,  5  roubles  =  9  marcs  of  Hamburg;  3 
marcs  =  1  Spanish  dollar;  and  9  dollars  =  50  francs. 

We  write  the  quantities  which 
are  equivalent  to  each  other, 
as  antecedent  and  consequent, 
making  each  consequent  of  the  same 
denomination  as  the  next  antece 
dent.  The  like  factors  on  op 
posite  sides  are  cancelled,  and 
the  products  divided  as  in  $71, 
to  obtain  the  answer. 


£13  =  ^#5  roubles, 
rou.         £  =  0       marcs, 
marcs     fi  =  1        dollar, 
dol.          0  =  50     francs, 
francs  —  =  1500£ 
13)     375000 


2884Cfr.  15T53C. 


£13 
rou.  5 
mar.  3 

$     9 


1500£ 


/5  rou 
9  mar. 

\l$ 


The  question  may  be  otherwise  stated  in 
the  following  manner:  If  £13  produce  75 
roubles,  5  roubles  produce  9  marcs,  .3  marcs 
produce  §1.00,  and  $9. 00  produce  50  francs, 
how  many  francs  will  £1500  produce?  The 
second  set,  or  set  of  demand,  contains  but 
a  single  cause  and  effect.  The  first,  or 
given  set,  contains  a  number  of  causes  and 
effects,  but  they  are  so  connected,  that  all 

the  terms  may  be  multiplied  together,  as  a  single  compound 
term.  Thus,  if  £13  produce  75  roubles,  and  5  roubles  produce  9 
marcs,  £13  will  produce  -755-  of  9  marcs,  and  £13  X  5  will  produce 
75  X  9  marcs.  In  the  same  way,  it  may  be  shown  that  £13  X  5 
X  3  =  $75  X  9  X  1,  and  £13  X  5  X  3  X  9  =  75  X  9  X  1  X  50  fr. 
Then  how  many  francs  will  £1500  produce? 


fr. 


EXAMPLES  FOR  THE  PUPIL. 

1.  A  London  merchant  wishing  to  pay  1000  milrees  in 
Lisbon,  remits  as  follows :  To  Amsterdam,  at  36  schillings 
7  groats  per  £;  thence  to  Cadiz,  at  17  groats  for  2  rials  of 


§73.]  ARBITRATION    OF   EXCHANGE.  257 

plate;  thence  to  Leghorn,  at  17  pezze  for  100  rials;  thence 
to  Lisbon,  at  1497  rees  for  2  pezze.  How  many  pounds 
did  he  remit?  Ans.  £152  3s.  3  id. 

2.  If  a  merchant  of  New  York  remits  $5000  to  Havre, 
at  5fr.  35c.  for  $1 ;  thence  to  London,  at  49fr.  for  £2 ; 
thence  to  Hamburg,  at  1  marc  for  Is.  6d. ;  and  thence  to 
St.  Petersburg,  at  8  roubles  for  17  marcs,  how  many  rou 
bles  can  he  pay  with  his  remittance  ?* 

Ans.  6850rou.  74cop. 

3.  If  33  copecks  are  equal  to  5  English  pence,  11  Eng 
lish  pence  are  equal  to  3  piastres,  13  piastres  are  equal  to 
1  florin,  and  5  florins  are  equal  to  29  francs,  how  many 
francs  are  equal  to  9000  copecks  ?  Ans.  165f.  92c. 

4.  If  a  man  receives  $30  for  building  8  rods  of  wall,  and 
he  can  purchase  3  barrels  of  flour  for  $14,  and  3cwt.  of 
sugar  for  4  barrels  of  flour,  and  21  Ib.  of  tea  for  2cwt.  of 
sugar,  how  many  pounds  of  tea  could  he  purchase  by  build 
ing  17  rods  of  wall?  Ans.  107 Ib.  9£oz. 

5.  If  13  days'  work  will  purchase  1  hogshead  of  molas 
ses,  and  2  hogsheads  of  molasses  are  worth  5  tons  of  hay, 
and  3  tons  of  hay  are  worth  4  bags  of  coffee,  how  many 
bags  of  coffee  can  be  bought  with  39  days'  labor  ? 

Ans.  10  bags. 

6.  If  70  braces  of  Venice  are  equal  to  75  braces  of  Leg 
horn,  and  7  braces  of  Leghorn  are  equal  to  4  yards,  how 
many  yards  are  there  in  79.375  braces  of  Venice? 

Ans.  48jJ|yd. 

7.  A  merchant  in  New  York  orders  £500  sterling,  due 
him  in  London,  to  be  sent  by  the  following  circuit :  To 
Hamburg,  at  15  marcs  banco  per  £ ;  thence  to  Copenhagen, 
at  100  marcs  banco  for  33  rix-dollars ;  thence  to  Bordeaux, 


•  The  pupil  must  carefully  observe  the  rule,  and  make  each  conse 
quent  of  the  same  denomination  as  the  next  antecedent. 

17 


258  THE   COUNTING-HOUSE.  [ART.  XV. 

at  3  rix-dollars  for  18  francs;  thence  to  Lisbon,  at  125 
francs  for  18  milrees;  and  thence  to  New  York  at  $1.25 
per  milree.  What  was  the  arbitrated  value  of  a  dollar  by 
this  remittance  ?a  Ans.  3s.  8.89  +  d. 

8.  Amsterdam  exchanges  with  London,  at  34  schillings 
4  pfennings  per  £,  and  with  Lisbon  at  52  pfennings  for 
400  reas.     What  is  the  arbitrated  exchange  between  Lon 
don  and  Lisbon,  by  way  of  Amsterdam  ? 

Ans.  £1  =  30169^3. 

9.  The  exchange  between  New  York  and  London  is  $4.84 
per  pound;  between  London  and  Amsterdam,  35  schillings 
per  pound  ;  between  Amsterdam  and  Paris,  58  groats  for 
6  francs;  between  Paris  and  Venice,  10  francs  per  ducat; 
and  between  Venice  and  Cadiz,  360  maravedis  per  ducat. 
How  many  maravedis  will  be  equivalent  to  $4500,  by  this 
circuitous  remittance  ?  Ans.  1454260  JjjjjjJ  mar. 

10.  If  100  Ib.  of  Amsterdam  are  equal  to  105  Ib.  of  Ant 
werp,  100  Ib.  of  Antwerp  to  142  Ib.  of  Genoa,   100  Ib.  of 
Genoa  to  70  Ib.  of  Leipsic,  and  100  Ib.  of  Leipsic  to  104  Ib. 
of  America,  how  many  Ib.   of  Amsterdam  are  equal  to 
1491  Ib.  of  America?  Ans.  1373  j»  Jib. 


ALLIGATION. 

When  it  is  desired  to  make  a  mixture  of  a  given  value, 
with  a  variety  of  ingredients,  the  following  method  is  usu 
ally  adopted  :  — 

1.  Having  written  the  values  of  the  ingredients  in  a  perpen 
dicular  column,  connect  by  a  line  each  value  that  is  less 
than  the  required  average  with  one  or  more  that  is  greater, 
and  each  value  that  is  greater  with  one  or  more  that  is  less. 

2.  Write  the  difference  between  each  value  and  the  average, 


a  After  finding  the  number  of  dollars  which  are  equivalent  to  .£500, 
the  Arbitrated  value  of  $1  is  found  by  dividing  £500  by  that  number. 


§  74.]  ALLIGATION.  259 

opposite  the  ingredient  with  which  that  value  is  connected, 
and  the  difference,  (or  the  sum  of  the  differences,  if  there 
be  more  than  one,)  opposite  each  ingredient,  will  be  the  quan 
tity  of  that  ingredient  required. 

EXAMPLES  FOR  ILLUSTRATION. 

1.  How  much  sugar  at  5cts.,  7cts.,  Sets.,  lOcts.,  and  12cts.,  must 
be  mixed  together,  that  the  mixture  may  be  worth  9cts.  a  pound  ? 


1st  Ans.  4  Ib.  at  5,  1  Ib.  at  7,  4  Ib.  I  2d  Ans.  3  Ib.  at  5,  31b.  at  7,  1  Ib.  i  3d  Ans.  1  Ib.  at  5,  3  Ib.  at  7,  3Ib 
it  8,  7  Ib.  at  10,  5  Ib.  at  12.  I  at  8,  1  Ib.  at  10,  6  Ib.  at  12.  |  at  8,  4  Ib.  at  10,  3  Ib.  at  12. 

We  may  obtain  as  many  answers  as  there  are  different  ways  of 
connecting  the  numbers  above,  with  those  below  the  average. 

To  prove  the  rule  correct,  let  us  examine  the  second  of  the 
above  answers.  If  we  were  mixing  sugars  at  5  and  12cts.  to  sell 
the  mixture  at  9cts.,  we  should  gain  4cts.  on  every  pound  of  the 
former,  and  lose  Sets,  on  every  pound  of  the  latter.  Then,  on  3  Ib. 
of  the  former  we  should  gain  12cts.  and  on  41b.  of  the  latter  we 
should  lose  12cts. ;  therefore,  if  we  mix  these  quantities,  we  shall 
neither  gain  nor  lose  by  selling  the  mixture  at  9cts.  In  the  same 
way  it  may  be  shown  that  31b.  at  7cts.  and  21b.  at  12cts.,  lib. 
at  Sets,  and  1  Ib.  at  lOcts.  may  be  sold  at  the  average  of  9cts., 
and  the  same  reasoning  will  prove  the  truth  of  each  of  the  other 
answers. 

2.  A  farmer  wishes  to  mix  10  bushels  of  barley  at  50cts.,  4bushels 
of  oats  at  40cts.,  and  16  bushels  of  rye  at  75cts.  with  wheat  at 
$1.25,  and  corn  at  OOcts.  a  bushel,  so  that  the  mixture  may  be 
worth  $1.00  per  bushel. 

We  may  regard  the  limited  quantities  as  a  single  ingredient  of 
30  bushels,  worth  62cts.  a  bushel.  Proceeding  in  the  usual  way, 

we  find  that  25  bushels  at  62cts.,  25  at  (     62 25 

25 

38  +  10 


90cts.,  and  48  at  $1  25,  would  give  us     1.00  \     90— 
a  mixture  of  the  desired  average  value. 


But  as  we  have  30  bushels  at  62cts.,  we  must  take  4  9  or  4  of 

>*  o  5 

these  proportionate  quantities,  and  we  have  30  bushels  at  90cts., 
and  57|bu.  at  $1.25,  for  the  answer. 


260  THE   COUNTING-HOUSE.  [ART.  XV. 

In  most  questions  in  Alligation,  an  infinite  number  of 
answers  may  be  obtained,  but  it  will  readily  be  perceived 
that  the  preceding  method  gives  only  a  few  of  those  answers. 
The  following  rule  is  not  only  more  general,  but  also  more 
analytical  in  its  character. 

Assume  any  quantity  you  please  of  each  ingredient,  find 
the  cost  of  the  whole,  and  also  the  cost  of  the  same  quan 
tity  at  the  mean  rate  proposed.  If  the  assumed  quantities 
cost  TOO  MUCH,  take  such  additional  quantities  of  the  lower 
priced,  or  such  diminished  quantities  of  the  higher  priced 
ingredients,  as  will  exactly  counterbalance  the  excess.  If 
TOO  LITTLE,  take  such  additional  quantities  of  the  higher 
priced,  or  such  diminished  quantities  of  the  lower  priced 
ingredients,  as  will  exactly  counterbalance  the  deficiency. 

EXAMPLE  FOR  ILLUSTRATION. 

In  what  proportion  should  I  mix  sugars  at  5cts.,  7cts.,  8cts., 
lOcts.,  and  12cts.,  in  order  that  the  mixture  may  be  worth  9cts. 
a  pound  ? 

We  may  commence  by  taking  any  3ib.  at  5cts.  =  .15 
quantity  we  please  of  each  ingredient,  j  ]b[  at  7ctg[  =  '07 
and  finding  the  cost  of  the  whole.  If,  21b.  at  Sets.  =  .16 
for  example,  we  take  3  Ib.  at  5cts.,  1  Ib.  41b.  at  lOcts.  ==  .40 
at  7cts.,  21b.  at  Sets.,  41b.  at  lOcts.,  and  51b.  at  12cts.  =  .60 
5  Ib.  at  12cts.,  (making  15  Ib.  in  all,)  the  15  Ib.  cost  1.38 
whole  cost  will  be  $1.38.  But  15  Ib.  at  15  Ib.  at  9cts.  =  1.35 
9cts.  would  cost  only  $1.35,  therefore  our  Excess  .03 

estimated  quantities  give  an  excess  of  3 

cents  above  the  required  cost.  To  balance  this  excess,  we  must 
either  add  some  of  the  sugar  that  costs  less  than  the  average  price 
proposed,  or  take  out  some  of  that  which  costs  more  than  the 
average.  For  every  pound  that  we  add  at  5  cents,  there  will  be 
a  deficiency  of  4  cents  from  the  mean  rate  ;  for  every  pound  at 
7cts.,  a  deficiency  of  2cts. ;  for  every  pound  at  Sets.,  a  deficiency 
of  let.  Then  we  may  either  add  f  of  a  Ib.  more  at  Sets.,  or  1  £lb. 
more  at  7cts.,  or  31b.  more  at  Sets.,  or  lib.  more  at  Sets,  and 
1  Ib.  at  7cts.,  or  any  other  quantity  that  will  make  a  deficiency 
of  Sets. 

If  the  quantities  first  assumed  had  fjiven  a  deficiency  instead  of 


§  74.]  ALLIGATION.  261 

an  excess,  we  should  have  been  obliged  to  take  some  additional 
quantity  of  one  or  more  of  the  ingredients  whose  value  is  greater 
than  the  proposed  average,  or  to  take  out  .  ~~ 

some  of  the  ingredients  whose  value  is  g  -^  ^  7cts'  __  ' %\ 
less  than  the  average.  The  cost  of  the  21b.  at  Sets.  =  .16 
quantities  assumed  in  the  margin,  would  1  lb.  at  lOcts.  =  .10 
be  only  91cts.,  but  12  lb.  at  9cts.  would  2  lb-  at  12cts-  = 
cost  $1.08.  There  is,  therefore,  a  defi-  12  lb.  cost  .91 
ciency  of  17cts.,  which  may  be  balanced  12  lb.  at  9cts.  =  1.08 
by  taking  enough  of  the  sugar  that  costs  Deficiency  .17 

more  than  9cts.,  to  make  an  excess  of 

17cts.  Thus,  5  lb.  additional  at  12cts.,  and  2  lb.  at  lOcts.,  would 
answer  the  required  conditions.  The  pupil  may  determine  other 
values  for  himself. 

The  following  are  therefore  some  of  the  answers  to  the  proposed 
question:— (1.)  3f  lb.  at  5cts.,  lib.  at  7cts.,  21b.  at  Sets.,  41b.  at 
lOcts.,  and  51b.  at  12cts.  (2.)  31b.  at  Sets.,  2Jlb.  at  7cts.,  2  lb. 
at  Sets.,  41b.  atlOcts.,  and  51b.  at  12cts.  (3.)  31b.  at5cts.,  1  lb. 
at  7cts.,  51b.  at8cts.,  4  lb.  at  lOcts.,  and  51b.  at  12cts.  (4.)  31b. 
at  5cts.,  21b.  at  7cts.,  31b.  at  8cts.,  41b.  at  lOcts.,  51b.  at  12cts. 
(5.)  41b.  at  5cts.,  31b.  at  7cts.,  21b.  at  Sets.,  31b.  at  lOcts.,  71b. 
at  12cts. 

Let  the  pupil  perform  the  above  examples  by  taking  out  some  of 
the  quantities  originally  assumed. 

EXAMPLES  FOR  THE  PUPIL. 

1.  A  mixture  is  made  of  24bu.  of  grain  at  $1.10  per  bu., 
28bu.  at  $.90,  and  44bu.  at  $.60.     How  must  the  mixture 
be  sold,  in  order  to  gain  $T3^  per  bushel  ?a 

Ans.  $1  per  bushel. 

2.  What  is  the  fineness  of  a  composition,  consisting  of 
31b.  6oz.  of  gold,  23  carats  fine,  41b.  8oz.  of  21  carats, 
3  lb.  9oz.  of  20  carats,  and  21b.  2oz.  of  alloy? 

Ans.  18  carats. 

3.  A  grocer  mixes  65  lb.  of  sugar  at  $.08,  30  lb.  at  $.07, 

a  The  first  three  examples  are  not,  strictly  speaking,  examples  in 
Alligation,  but,  in  accordance  with  general  custom,  they  are  insertect 
in  this  place.  See  Remark  at  the  close  of  Section  68. 


262  THE   COUNTING-HOUSE.  [ART.  XV. 

25  lb.  at  $.05,  and  20  Ib.  at  $.06  J.     How  must  the  mixture 
be  sold,  in  order  to  gain  25  per  cent.  ? 

Ans.  Sfcts.  per  lb. 

4.  In  what  proportions  may  I  mix  teas  at  60cts.,  70cts., 
$1.10,  and  $1.20,  per  pound,  so  as  to  gain  12  £  per  cent. 
by  selling  at  $.90  per  pound?  a         Ans.  4  lb.  at  60  ;  3  lb. 

at  70;  lib.  at  $1.10;  21b.  at  $1.20. 

5.  A  corn  merchant  bought  wheat  at  $1.20,  at  $1.10,  at 
$.90,   and  at  $.70   per  bushel;  but  the  markets   having 
fallen,  he  is  desirous  to  sell  at  $.80  per  bushel,  and  is  will 
ing  to  lose  20  per  cent.     In  what  proportions  may  a  mixture 
be  made,  to  answer  the  conditions  of  the  question  ? 

Ans.    3bu.  at  $1.20,  Ibu.  at  $1.10,  Ibu.  at  $.90, 
2bu.  at  $.70. 

6.  A  cask  contains  50  gallons  of  acid,  worth  70  cents 
per  gallon  ;  how  much  will  it  contain  after  the  value  of  the 
liquor  has  been  reduced  to  60  cents  per  gallon,  by  pouring 
in  water?  Ans.  5  8  i  gallons. 

7.  A  silversmith  mixes  20oz.  of  silver,  at  5s.  9d.  per 
ounce,  with  two  other  kinds,  at  5s.  6d.,  and  5s.  3d.,  and  as 
much  alloy  as  reduces  the  mass  to  the  value  of  4s.  lOd. 
per  ounce.    Required  the  weight  of  the  whole  composition. 

Ans. 


8.  A  refiner  melts  14  lb.  of  gold,   22  carats  fine,  with 
16  lb.  of  20  carats  fine.     How  much  alloy  must  be  added, 
in  order  to  make  the  composition  18  carats  fine  ? 

Ans.  4|lb. 

9.  A  New  York  merchant  shipped  8000  bushels  of  grain, 
consisting  of  wheat,  barley,  and  rye,  which  he  sold  in  Lon 
don  at  7s.  4d.  per  bushel.     The  prime  cost  of  the  white 
wheat  was  6s.  6d.,  the  red  wheat  5s.  9d.,  the  barley  3s.  4d., 
and  the  rye  4s.  3d.  per  bushel.     The  port  charges  and  other 

*  An  infinite  number  of  answers  can  be  obtained  to  some  of  the 
lemaining  examples  in  this  section,  besides  the  ones  that  are  given. 


§75.]  GENERAL  AVERAGE.  263 

incidental  expenses,  amounted  to  £58  6s.  8d. ;  the  freight 
was  lOd.  per  bushel,  and  he  gained  9 id.  per  bushel  by  the 
transaction.  How  many  bushels  of  each  kind  of  grain 
would  answer  the  conditions  of  the  question  ? 

Ans.  2285|bu.  white  wheat;   3857^bu.  red  wheat; 
285fbu.  barley;  1571-f-bu.  rye. 

75.    GENERAL  AVERAGE.* 

Whenever  any  sacrifice  of  property  is  made,  or  any  ex 
pense  necessarily  incurred,  for  the  preservation  of  a  ship  and 
cargo,  the  loss  is  divided  among  all  the  parties  interested, 
either  as  owners  of  the  vessel,  or  of  the  property  on  board. 

"  Thus,  where  the  goods  of  a  particular  merchant  are  thrown 
overboard  in  a  storm,  to  save  the  ship  from  sinking ;  or  where 
the  masts,  cables,  anchors,  or  other  furniture  of  the  ship  are  cut 
away  or  destroyed  for  the  preservation  of  the  whole ;  or  money 
or  goods  are  given  as  a  composition  to  pirates  to  save  the  rest ;  or 
an  expense  is  incurred  in  reclaiming  the  ship,  or  defending  a  suit 
in  a  foreign  court  of  admiralty,  and  obtaining  her  discharge  from 
an  unjust  capture  or  detention  ;  in  these  and  the  like  cases,  where 
any  sacrifice  is  deliberately  and  voluntarily  made,  or  any  expense 
fairly  and  bond  fide  incurred,  to  prevent  a  total  loss,  such  sacrifice 
or  expense  is  the  proper  subject  of  a  general  contribution,  and 
ought  to  be  ratably  borne  by  the  owners  of  the  ship,  freight, 
and  cargo,  so  that  the  loss  may  fall  equally  on  all,  according  to 
the  equitable  maxim  of  the  civil  law: — No  one  ought  to  be  enriched 
by  another's  loss."b 

The  loss  is  distributed  in  such  cases,  by  GENERAL  AVER 
AGE.  But  if  any  sacrifice  is  made  for  the  sake  of  the  ship 
only,  or  of  the  cargo  only,  the  loss  must  be  borne  by  the 
parties  immediately  interested,  and  is  consequently  defrayed 
by  a  particular  average. 

In  New  York,  only  J  of  the  value  of  the  freight  is  con 
tributory  ;  in  the  other  United  States  ports,  f  of  the  value 
Ls  taken.  The  remainder  of  the  freight  is  reserved  for  sea 
men's  wages,  because  if  the  seamen  were  laid  under  this 

«•  McCulloch,  Hunt's  Mer.  Mag.  &  Mr.  Serjeant  Marshall. 


264 


THE   COUNTING-HOUSE. 


[ART.  XV. 


obligation,  they  might   be  tempted  to  oppose   a  sacrifice 
necessary  for  the  general  safety. 

When  the  loss  of  masts,  or  ship  furniture,  is  compensated 
by  general  average,  as  the  new  articles  are  supposed  to  be 
of  more  value  than  those  that  were  lost,  only  f  of  the  value 
is  contributed. 


EXAMPLES. 

1.  It  became  necessary,  in  the  Downs,  to  cut  the  cable  of 
a  ship  destined  for  Hull ;  the  ship  afterwards  struck  on  a 
bar,  and  the  captain  was  compelled  to  cut  away  the  mast, 
and  throw  overboard  part  of  the  cargo ;  in  which  operation 
another  part  was  injured.  The  ship,  being  cleared  from  the 
sands,  was  forced  to  take  refuge  in  Ramsgate  harbor,  to 
avoid  further  injury  from  the  storm.  Required,  from  the 
following  statement,  the  proper  adjustment  of  the  loss. 


AMOUNT    OP   LOSSES. 

Goods  of  A.,  cast  over 
board    .    .    .  $2000 


800 
400 

800 


700 

16 

4 


CONTRIBUTORY   VALUES. 

Goods  of  A.,   cast  over 
board          .         .         .  §2000 

Sound  value  of  B.'s  goods    4000 

Goods  of  C.  . 
«  "  D.  . 
"  "  E.  . 

Value  of  the  ship  . 

|  of  freight  . 


2000 
8000 
20000 
8000 
3200 


Damage  of  B.'s  goods     . 
Freight  of  A.'s  goods 
§  of  new  cable,  anchor, 

and  mast     . 
Pilotage,  port-duties,  and 

expenses    of    bringing 

ship  off  the  sands 
Adjusting  average 
Postage 

Total  loss 

Ans.  The  owners  of  the  vessel  receive  $800 ;  C.  pays  $200. 

A.  receives     .         .         .        $1800;  D.    «    $800. 

B.  "  ...          $400;  E.    "  $2000. 

2.  In  a  storm,  goods  belonging  to  A.,  worth  $500,  were 
thrown  overboard,  and  the  losses  of  the  owners  of  the  vessel 
amounted  to  $1500.  Adjust  the  general  average,  the  total 


§4720    Tot.  contributory  values  $47200 


§76.] 


PRODUCT   OF   MINES. 


265 


value  of  A.'s  goods  being  $800;  B.'s  goods,  $1200;  C.'s 
goods,  $3000;  value  of  the  ship,  $10000;  total  freight, 
$1500.  Ans.  A.  receives  $400 ;  B.  pays  $150. 

Owners  receive  $125 ;  C.  pays  $375. 


XVI.    STATISTICS. 


PRODUCT  OF  MINES  IN  THE  UNITED  STATES,  ACCORD- 
ING  TO  THE  CENSUS  OF  1840.a 


States  and  Territo 
ries. 

Cast  &  Kar. 
Tons. 

Lead. 
JE'ounds. 

Gold. 
Value. 

Other 

Metals 
Value. 

Coal. 

Tons  of  28 
bushels. 

Salt. 
Bushels. 

Granite, 
Marble.&c. 
Value. 

Maine  

6122 

$1600 

50000 

$107506 

N.  Hampshire.. 

1445 

1000 

10300 

1069 

1°00 

16038 

Massachusetts. 

15336 

2500 

376596 

790R55 

Rhode  Island.. 

4126 

1000 

17800 

Connecticut  .  .  . 

10118 

1357 

1500 

313469 

Vermont    

7398 

70500 

33855 

New  York  
New  Jersey  .  .  . 

82781 
18285 

670000 

84564 
39.550 

2867884 
500 

1541480 
35721 

Pennsylvania.  . 

185639 

100200 

1274709 

549478 

238831 

Delaware  

466 

1160 

16000 

Maryland  

16776 
24697 

878618 

"551758 

28800 

7929 
379569 

1200 
1745618 

22750 
84489 

N.  Carolina  
South  Carolina 

1931 
2415 
494 

10000 

25-5618 
37418 
121881 

1000 

-3 

4493 
2250 

3350 
3000 
51990 

Alabama  

105 

61230 

'"845 

13700 

Mississippi  .... 

Louisiana  

2766 

Tennessee  
Kentucky  

25802 
32813 



1500 

498 
23131 

219695 

30100 
19592 

Ohio  

42702 

16000 

125775 

297350 

195831 

Indiana  

830 

8614 

6100 

35021 

Illinois  
Missouri  

158 
298 

8755000 
5295455 

200 

15600 

152S2 

8904 

20000 
13150 

74228 

OQUO 

Arkansas  

196 

8700 

15500 

Michigan  

601 

2700 

Florida 

120001 

2650 

Wisconsin  

3 

15129350 

968 

Iowa  

500000 

357 

350 



Total, 

EXAMPLES. 

1.  Find  the  total  products  embraced  in  the  above  table. 

2.  Estimating  the  average  value  of  the  iron  at  $22  per 
ton,  lead  at  4J  cents  per  lb.,  coal  at  $3.75  per  ton,  and  salt 
at  $.40  per  bushel,  what  was  the  entire  value  of  the  above 
products  ? 

a  Tucker's  Progress. 


266 


STATISTICS. 


[ART.  XVI. 


TABLE    OF    THE    AGRICULTURAL    PRODUCTS    OF    THE 
UNITED    STATES,    ACCORDING    TO    THE    CENSUS    OF 

1840. a 


States  and  Territo 
ries. 

Bushels  of 
Wheat. 

Bushels  of 
harley. 

Bushels  of 
Oats. 

Bushels  of 
Rye. 

Bushels  of 
Buckwheat 

Bu?heh  of 
Ind.  Corn. 

Jiushrls  of 
Potaioes. 

10392280 
6206606 
5385652 
91  1973 
3414238 
8869751 
30123614 
2072069 
9535663 
200712 
103643-3 
2944660 
2609239 
2698313 
1291366 
1708356 
1630100 
831341 
1901370 
1055085 
5605021 
1525794 
2025520 
783768 
293608 
2109205 
264617 
419608 
234063 
12035 

848166 
422124 
1579-23 
3098 
87009 
495800 
12286418 
774203 
13213077 
315165 
33457S3 
10109716 
1960855 
968354 
1801830 
828052 
1966-26 
60 
4569692 
4803152 
16571661 
4049375 
3335303 
1037386 
105878 
2157108 
412 
212116 
154693 
12147 

355161 
121899 
165319 
66490 
33759 
54781 
2520068 
12501 
209893 
5260 
3594 
87430 
3574 
3967 
12979 
7692 
1654 

4809 
17491 
212440 
28015 
822ol 
9801 
760 
127802 
30 
11062 
728 
294 

1076409 

1296114 
1319680 
171517 
145326-2 
2222584 
20675847 
3083524 
20641819 
927405 
353421  1 
13451062 
3193941 
1186208 
1610030 
1406353 
668624 
107353 
7035678 
7155974 
14393103 
5981605 
4988008 
2231947 
189553 
2114051 
13829 
406514 
216385 
15751 

137941 
308148 
536014 
34521 
737424 
230993 
2979323 
1665820 
6613873 
33546 
723577 
1482799 
213971 
44738 
60693 
51008 
11444 
1812 
301320 
1321373 
814205 
1296-21 
B8197 
68608 
6-219 
34236 
305 
1965 
3792 
5081 

51543 
105103 
87000 
2979 
303043 
228116 
2287885 
856117 
2113742 
11299 
73606 
243822 
15391 
72 
141 
58 
61 

950.328 
1162572 
1809192 
450498 
1500441 
1119678 
10972286 
4361975 
14240022 
2099359 
8233086 
34577591 
23893763 
14722805 
20905122 
20947004 
13161237 
5952912 
44(^6188 
:WH7  120 
33668144 
28155887 
22634211 
17332524 
4846632 
2277039 
898974 
379359 
1406-211 
39485 

N.  Hampshire.. 
Massachusetts 
Rhode  Island.. 
Connecticut.  .  • 

New  York  .... 
New  Jersey.  .  . 
Pennsylvania.  . 
Delaware  
Maryland  

N.  Carolina.  .. 
S.  Carolina.... 

Alabama  
Mississippi..  .. 
Louisiana  
Tennessee...... 
Kentucky  
Ohio 

17118 
8169 
633139 
49019 
57884 
15318 
88 
113592 

10654 
6212 
272 

Illinois  
Missouri  
Arkansas  
Michigan  
Florida  
Wisconsin  
Iowa  
Dist.of  Col.  ... 

Total  

EXAMPLES. 

1.  Find  the  total  products  of  each  article  embraced  in  the 
above  table. 

2.  Find  the  value  of  the  entire  product  of  the  wheat,  at 
$1.10  per  bushel;  of  the  barley,  at  $.65  per  bu.;  of  the 
oats,  at  $.40  per  bu.;  of  the  rye,  at  $.70  per  bu.;  of  the 
buckwheat,  at  $.75  per  bu.;  of  the  corn,  at  $.80  per  bu.; 
of  the  potatoes,  at  $.37£  per  bu. 

3.  What  percentage   of  the  entire  wheat  harvest  was 
produced  in  1840,   by  the  four  principal  wheat-growing 

States  ? 


Tucker's  Progress. 


§78.] 


AGRICULTURAL   PRODUCTS. 


267 


7" 8.       TABLE  OF  AGRICULTURAL  PRODUCTS.— CONTINUED.* 


States,  &c.   j  you*. 

Tobacco. 
Pounds. 

Pounds. 

Cotton. 
Pounds. 

Wool. 
Pounds. 

Sugar. 
Pounds. 

Dairy 
Products. 

Maine  
N.Hamp.... 
Mass  
R.  Island... 
Connecticut 
Vermont..  .  . 
New  York  . 
New  Jersey 
Pennsylv'a. 
Delaware  .. 
Maryland..  . 
Virginia...  . 
N.  Carolina 
S.  Carolina. 
Georgia  
Alabama...  . 
Mississippi. 
Louisiana... 
Tennessee.. 
Kentucky... 
Ohio 

691358 
496107 
56939,3 
63149 
426701 
836739 
3127017 
334861. 
1311613 
2-2  [S3 
1066371 
3617081 
101369 
21618 
16969^ 
12718 
171 
24651 
31233 
88306 
1022037 
178029 
164932 
49083 
586 
130805 
1197 
30938 
17953 
1331 

30 
115 
64955 
317 
471657 

1465551 
1260517 
941906 
183830 
889870 
3699235 
9815295 
397207 
3048564 
61404 
488201 
2538374 
625014 
299170 
371303 
220353 
175196 
49283 
1060332 
1786847 
3685315 
1237919 
650007 
562265 
61943 
153375 
7285 
6777 
23039 
<U7  ..    ... 

257461 
1162368 
579227 
50 
51764 
4647934 
10018109 
56 
2265755 

$1496902 
1638513 
2373299 
223229 
1376534 
2008737 
10196021 
1328032 
3187292 
113828 
457466 
1480488 
674319 
577810 
605172 
265200 
359585 
153069 
472141 
931363 
1848869 
742269 
428175 
100132 
59205 
301052 
23094 
35677 
23609 
5566 



585 
744 
1922 
325018 
272 
24816012 
75317106 
16772359 
51519 
162894 
273302 
83471 
119821 
29550432 
53436909 
5912275 
1820306 
564326 
9067913 
148139 
1602 



'"  "" 

334 
5673 
3494483 
5192tfi90 
61710274 
163392396 
117138823 
193401577 
152555368 
27701277 
691456 

"  ISO 
200947 
121122 
6028612 

36266 
154  1833 
7163 
30000 
329744 
10143 
77 
119947720 
258073 
1377835 
6363386 
3727795 
399813 
274853 
1542 
1329784 
275317 
135288 
41450 

2956 
2820388 
60590861 
12384732 
149019 
777195 
3604534 
7977 
16376 

Indiana  
Illinois  
Missouri  .  .  . 
Arkansas.  . 
Michigan.  ... 
Florida  
Wisconsin.. 
Iowa  
Dist.  of  Col. 

Total  ..." 

460 
50 
5454 

75274 
115 
8076 
55550 

481420 

12110533 

1 

EXAMPLES. 

1.  Which  of  the  States  yielded  the  heaviest  product  of 
each  of  the  foregoing  articles,  and  what  percentage  of  the 
entire  product  of  each  article,  was  the  heaviest  product  'I 

2.  Find  the  average  amount  yielded  by  each  of  the  pro 
ducing  States,  of  each  article. 

3.  Find  the  total  product  of  each  article,  and  the  value 
of  the  cotton,  at  12  ?  cents  per  pound. 

4.  Find  what  percentage  of  each  of  the  foregoing  articles 
was  produced  by  the  New  England  States ;  by  the  Middle 
States;    by  the    Southern    States;    by  the   South-western 
States;  by  the  North-western  States. 

»  Tucker's  Progress. 


268 


STATISTICS. 


[ART.  XVI. 


70         TABLE    OF    OCCUPATIONS    IN    THE    UNITED    STATES, 
ACCORDING    TO  THE  CENSUS  OF  1840.a 


States  and  Territo- 
ries. 

Mining. 

Agncul- 
ture. 

Com- 

Manu 
factures. 

Ocean 
Navig. 

Internal 
Navig. 

Learned 

Total. 

36 

101630 

29°  1 

21879 

10091 

539 

1889 

N.  Hampshire.. 

13 

77949 
73150 

1379 
1303 

17826 
13174 

452 
41 

198 

146 

1640 
1563 

Massachusetts. 
Rhode  Island.. 
Connecticut...  . 

499 
35 
151 

87837 
16617 
56955 

8063 
1348 
2743 

85176 
21271 
27932 

27153 
1717 
2700 

372 

228 
431 

3801 
457 
1697 

N  En°"  States 

67508° 

New  York  
New  Jersey  — 
Pennsylvania.. 
Delaware  
Maryland  
Dist  of  Col 

1898 
•2(30 
4603 
5 
320 

455954 
56701 
207533 
16015 
72016 
384 

28468 
2283 
15338 
467 
3281 
240 

173193 

27004 
105883 
4060 
21529 
2278 

5511 
1143 
1815 
401 
717 
1°6 

10167 
1625 
3951 
235 
1528 
80 

14111 
1627 
6706 
199 

1666 
203 

Middle  States 

1°51560 

Virginia  

1995 

31b'771 

6361 

54147 

582 

2952 

3866 

N.  Carolina  
South  Carolina, 
Georgia  - 

589 
51 
574 
1 

217095 
[98363 

20<i:jSJ 
12117 

1734 

1958 
2428 

481 

14322 
10325 
7984 

1177 

327 
381 
262 
435 

379 
348 
352 

118 

1086 
1481 
1250 
204 

1073879 

Alabama  
Mississippi..  .. 
Louisiana  
Arkansas  
Tennessee  

96 
14 
1 
41 
103 

177439 
139724 
79289 
26355 

227739 

2212 
1303 
8549 
215 
2217 

7195 
4151 
7565 
1173 
17815 

256 
33 
1322 
3 
55 

758 
100 
662 
39 
302 

1514 
1506 
1018 
301 
2042 

Southw  States 

713107 

Missouri  ., 
Kentucky  

Ohio 

742 
331 

704 

92408 
197738 
070579 

2522 
3448 
9201 

11100 
23217 
66°65 

39 
44 
212 

1885 
968 
3323 

1469 

2487 
5663 

Indiana  
Illinois  
Michigan  
AVisconsin  .... 

233 

782 
40 
794 
217 

148806 
105337 
56521 
7047 
10469 

3076 
2506 
728 
479 
355 

20590 
13185 
6890 
1814 
1629 

89 
63 
24 
14 
13 

627 

310 
166 

209 

78 

2257 
2021 
904 
259 
365 

Northw  States 

1085242 

Total  

15211 

3719951 

117607 

791749 

56021 

33076 

65255 

EXAMPLES. 

1.  Fill  all  the  blanks  in  the  above  table. 

2.  Find  the  average  number  employed  in  each  occupa 
tion,  in  each  of  the  New  England  States ;  in  each  of  the 
Middle  States ;  in  each  of  the  Southern  States  \  in  each  of 
the  South-western  States;   in  each  of  the  North-western 
States ;  in  the  entire  Union. 


a  Tucker's  Progress. 


§80.] 


EDUCATION. 


269 


TABLE  SHOWING  THE  STATE  OF  EDUCATION  IN  THE 
UNITED  STATES,  AND  THE  NUMBER  OF  WHITE  PER 
SONS  OVER  20  YEARS  OF  AGE  WHO  COULD  NOT  READ 
OR  WRITE,  ACCORDING  TO  THE  CENSUS  OF  lS40.a 


States  and  Territories. 

ll 
11 

Students. 

ji 

III 

111 

' 
C 

£>» 

l§ 

£1 

Scholars. 

III 

r° 

Illiterate. 

M'line 

4 

9(56 

86 

8477 

3385 

164477 

60°  12 

qn<4i 

New  Hampshire.  .  .  . 

2 
3 

433 
233 

68 
40 

5799 
4113 

2127 

2402 

83632 

8°817 

7715 
14701 

942 
2276 

Massachusetts  
Rhode  I^l'ind 

4 
2 

769 
3°4 

251 
5.) 

16746 
3664 

33(52 
434 

160257 
17355 

158351 

4448 

4 

83° 

127 

4865 

1619 

65739 

10912 

526 

New  England  States 

12 

1085 

505 

34715 

10593 

5O>367 

27075 

44452 

3 

443 

(j(j 

30°7 

1°07 

52583 

7128 

Pennsylvania  

20 
1 

2034 
23 

290 

20 

15970 
761 

4978 
152 

179989 
6904 

73908 
1571 

33940 
4832 

12 

813 

133 

4°89 

565 

16^51 

66'->4 

District  of  Columbia 

2 

224 

26 

1389 

29 

851 

482 

1033 

Middle  States 

Virginia 

13 

1007 

3so 

11083 

1561 

35331 

9791 

North  Carolina  
South  Carolina  

2 
1 
11 

158 
168 
(j.>2 

111 
117 

176 

4398 
4326 

7878 

632 
566 
601 

14937 
12520 
15561 

124 
3524 

56609 
20615 

Florida  

18 

732 

51 

925 

14 

-lonq 

Southern  States  

2 

150 

114 

5018 

639 

16°43 

7 

451 

71 

2553 

382 

8236 

107 

8360 

Louisiana  

12 

989 

52 

8 

1995 
300 

179 
113 

3573 
2614 

1190 

4861 
6567 

8 

492 

152 

5539 

983 

25090 

6907 

South-west'n  States 

Missouri  
Kentucky  

6 
10 

495 
1419 

47 
116 

1926 
4906 

612 

952 

16788 
24641 

526 
429 

19457 
40018 

Ohio  

18 

1717 

73 

4310 

5186 

218609 

51812 

35394 

Indiana  

4 
5 

322 
311 

54 
42 

2946 
1967 

1521 
1°41 

48189 
34876 

6929 

38100 

Michigan   

5 

158 

12 
2 

485 
65 

975 

29701 
1937 

998 
315 

2173 

25 

63 



North-west'n  States 

Total  

] 

EXAMPLE. 

1.  Fill  the  blanks  in  the  above  table,  and  find  the  per 
centage  of  the  entire  population1"  embraced  in  each  class,  in 
each  section  of  the  Union. 


Tucker's  Progress. 


b  See  Table  of  Population,  page  36. 


270 


STATISTICS. 


[ART.  XVI. 


e  |         SUMMARY  OF  THE  ANNUAL  PRODUCTS   OF  INDUSTRY 
IN    THE    SEVERAL   STATES,   AS   ESTIMATED   IN    THE 
CENSUS  OF  1840.a 


Staffs  an  1 
Territories. 

VALUE  OF  ANNUAL  PRODUCTS  FROM 

Agriculture. 

Manufac- 
tu  res. 

Commerce. 

Mining. 

Forest. 

Fisheries. 

Total. 

Maine.  .. 
N.Hamp 
Vermont 
Mass.  .  .  . 
R.  Island 
Connec't 

N.E.S. 

$158562/0 
11377752 
17879155 
16065627 

2199309 
11371776 

$5615303 
6545811 
5685425 
43518057 
8640626 
12778963 

$1505380 
1001533 
758899 
7004691 
1294956 
1963281 

8327376 
88373 
389488 
2020572 
162410 
820419 

$1877663 
449861 
430224 
377354 
44610 
181575 

$1280713 
92811 

'  '6483996 
659312 
907723 

74749889 

187657294 

N.York.. 
N.Jersey 
Pcnns'a.. 
Delaw'e. 
Maryland 
D.  of  Col. 

Middle  S. 

108275281 

162098,53 
68180924 
3198110 
17586720 
176912 

47454514 
10696257 
333.54279 
1538879 
6212677 
904526 

24311715 
1206929 
10593368 
266257 
34990J-7 
802725 

7408070 
1073<)21 
176(36  146 
54555 
1056210 

5040781 
361326 
120357& 
13119 
241194 

1316072 
124140 
35300 
181285 
225773 
87400 

390558303 

Virginia. 
N.  Caro'a 
S.  Caro'a 
Georgia.. 
Florida.  . 

S.  States 

Alabama 
Mississ'i 
Louisi'a. 
Arkans's. 
Tenness.. 

S.  W.  S. 

59085821 
26975831 
2155:3691 
31468271 
1834237 

8319218 
2053697 
2218915 
1953950 
434544 

5299451 
1322284 
2632421 

2248488 
464637 

3321629 
372486 
187608 
191631 
2700 

617760 
1446108 
549626 
117439 
27350 

95173 
251792 
1275 
584 
213219 

175321836 

2273267 
1453686 

7868898 
420635 
2239478 

:::•::::: 

24696513 
26194565 

22851375 

5086757 
31660180 

1732770 
1585790 
4087655 
1145309 
2477193 

81310 

"165280 
18225 
1371331 

177465 
205297 
71751 
217469 
225179 

' 

138607378 



Missouri 
Kentuc'y 
Ohio  
Indiana.. 
Illinois.. 
Michigan 
Wiscon. 
lo\va  .  .  . 

N.W.  S. 

10484263 

29226545 
37802001 
17247743 
1370146* 
4502889 
568105 
769295 

2360708 
5092353 
145&8091 
3676705 
3213981 
1376249 
304692 
179087 

2319245 
2580575 
8050316 
1866155 
119312,5 
622822 
189957 
136525 

187669 
1539919 
2142682 
660836 
293272 
56790 
384603 
13250 

448559 
184799 
1013063 
80000 
249811 
467540 
430530 
83949 

10525 
1192 

27663 

170989925: 





Total 

1063134736; 

EXAMPLE. 

1.  Fill  all  the  blanks  in  the  above  table,  and  find  what 
per  cent-age  of  the  total  products  was  derived  from  each 
source.  Find  also  the  proportion  to  each  inhabitant  of  the 
total  products  of  industry  in  each  State. 


a  Tucker's  Progress. 


§82.]  PERMUTATION.  271 

XVII.    PERMUTATION  AND  COMBINATION. 

83.     PERMUTATION. 

PERMUTATION  shows  the  number  of  changes  that  can  be 
made  in  the  order  of  a  given  number  of  things. 

PROBLEM  I. 

To  find  the  number  of  changes  that  can  be  made  of  any 
given  number  of  things,  all  different  from  each  other. 

How  many  changes  may  be  made  in  the  position  of  4  persons 
at  table  ? 

If  there  were  but  two  persons,  a  and  b,  they  could  sit  in  but 
two  positions,  ab  and  ba.  If  there  were  three,  the  third  could  sit 
at  the  head,  in  the  middle,  or  at  the  foot,  in  each  of  the  two 
changes,  and  there  could  then  be  1x2x3  =  6  changes.  If  there 
were  4,  the  fourth  could  sit  as  the  1st,  2d,  3d,  or  4th,  in  each  of 
these  6  changes,  and  there  would  then  be  1x2x3x4  =  24 
changes. 

RULE. 

Multiply  together  the  series  of  numbers  1,  2,  3,  &c.,  up  to  the 
given  number,  and  the  product  will  be  the  number  sought. 

1.  How  many  variations  may  be  made  in  the  order  of 
the  9  digits  ?  Am.  362880. 

2.  How  many  changes  may  be  made  in  the  position  of 
the  letters  of  the  alphabet  ? 

Am.  403291461126605635584000000. 

3.  How  long  a  time  will  be  required  for  8  persons  to  seat 
themselves  at  table  in  every  possible  order,  if  they  eat  3 
meals  a  day  ? 

PROBLEM  II. 

Any  number  of  different  things  being  given,  to  find  how 
many  changes  can  be  made  out  of  them  by  taking  a  given 
number  of  the  things  at  a  time. 

If  we  have  five  things,  each  one  of  the  5  may  be  placed  before 
each  of  the  others,  and  we  thus  have  5x4  permutations  of  2  out 


272  PERMUTATION    AND    COMBINATION.     [ART.  XVLT. 

of  5.  If  we  take  3  at  a  time,  the  third  thing  may  be  placed  as 
1st,  2d,  and  3d,  in  each  of  these  permutations,  and  we  have 
5x^x3  permutations  of  3  out  of  5.  For  a  similar  reason  we 
have  5  x  ^  X  3  X  ^  permutations  of  4  out  of  5,  £c. 

RULE. 

Take  a  series  of  numbers,  commencing  with  the  number  of 
things  given,  and  decreasing  by  1,  until  the  number  of  terms  is 
equal  to  the  number  of  things  to  be  taken  at  a  time.  The  product 
of  all  the  terms  will  be  the  answer  required. 

4.  How  many  changes  can  be  rung  with  8  bells,  taking 
5  at  a  time  ?  Ans.  6720. 

5.  How  many  numbers  of  4  different  figures  each,  can 
be  expressed  by  the  9  digits  ? 

6.  In  how  many  different  ways  may  10   letters   of  the 
alphabet  be  arranged  ?  Ans.  19275223968000. 

PROBLEM  III. 

To  find  the  number  of  permutations  in  any  given  number 
of  things,  among  which  there  are  several  of  a  kind. 

How  many  permutations  can  be  made  of  the  letters  in  the 
word  terrier? 

If  the  letters  were  all  different,  the  permutations,  according  to 
Problem  I.,  would  be  Ix2x3x4x5x6x  7  =  5040.  But 
the  permutations  of  the  three  r's  would,  if  they  were  all  different, 
be  1  X  -  X  3,  which  could  be  combined  with  each  of  the  other 
changes ;  the  number  must  therefore  be  divided  by  1  X  -  X  3. 
For  the  same  reason,  it  must  also  be  divided  by  1x2, 
on  account  of  the  2  e's.  Then  the  true  number  sought  is 
l_X_2x  3x4x5x6x7  =  ^Q 
1X2  X3X1X2 

RULE. 

Take  the  natural  series,  from  1  up  to  the  number  of  things  of 
the  first  kind,  and  the  same  series  up  to  the  number  of  things  of 
each  succeeding  kind,  and  form  the  continued  product  of  all  the 
series. 

By  the  continued  product,  divide  the  number  of  permutations 


§83.]  COMBINATION.  273 

of  which  the  given  things  would  be  capable,  if  they  were  all 
different,  and  the  quotient  will  be  the  number  sought. 

7.  How  many  changes  can  be  made  in  the  order  of  the 
letters,  in  the  word  Philadelphia?  Ans.  14968800. 

8.  How  many  different  numbers  can  be  made,  that  will 
employ  all  the  figures  in  the  number  119089907343  ? 

9.  How  many  permutations  can  be  made  with  the  letters 
in  the  word  Cincinnati  ?  Ans.  201600. 

83.    COMBINATION. 

COMBINATION  shows  in  how  many  ways  a  less  number 
of  things  may  be  chosen  from  a  greater. 

If  we  have  ten  articles,  each  may  be  combined  with  every  one 
of  the  nine  remaining  ones,  and  therefore  we  may  have  10  X  9 
permutations  of  2  out  of  10.  But  each  combination  will  evidently 
be  repeated ;  thus,  we  have  ab  and  ba,  ac  and  ca,  &c.  Therefore, 

10  v  9 
the  number  of  combinations  will  be  - — — 

2 

If  now  we  add  an  eleventh  article,  each  of  the  eleven  may  be 
joined  to  each  of  the  combinations  of  the  remaining  ten,  and  we 

shall  have  — ^ —  permutations.    But  each  combination  will 

be  three  times  repeated ;  thus  we  shall  have  abc,  bac,  and  cab  ; 
aid,  bad,  and  dab,  &c.     The  number  of  combinations  of  3  out  of 

11  will  therefore  be  n  X  10  X  ^      Hence  we  obtain  the  following 

RULE. 

Write  for  a  numerator  the  descending  series,  commencing  with 
the  number  from  which  the  combinations  are  to  be  made,  and  for 
a  denominator  the  ascending  series,  commencing  with  1,  giving  to 
each  series  as  many  terms  as  are  equivalent  to  the  number  in  one 
combination. 

Cancel  the  like  factors  in  the  numerator  and  denominator,  and 
divide. 

10.  How  many  combinations  of  4  letters,  can  be  made 
from  the  alphabet?  Ans.  14950. 

18 


274  INVOLUTION  AND   EVOLUTION.    [ART.  XVIII. 

11.  How  many  combinations  of  7  can  be  made  from  18 
apples?  Ans.  31824. 

12.  How  many  ranks  of  10  men,  may  be  made  in  a  com 
pany  of  80  ? 

13.  How  many  locks  of  different  wards,  may  be  unlocked 
with  a  key  of  6  wards  ?     [Find  the  number  of  combina 
tions  of  1,  2,  3,  4,  5,  and  6  in  6,  and  the  sum  of  all  the 
combinations  will  be  the  number  required.]        Ans.  63. 


XVIII.    INVOLUTION  AND  EVOLUTION. 

84.    INVOLUTION. 

INVOLUTION  is  the  repeated  multiplication  of  a  number 
by  itself. 

The  product  obtained  by  Involution  is  called  a  power. 
The  root  is  the  number  involved,  or  the  first  power.  If 
the  root  be  multiplied  by  itself,  or  employed  twice  as  a 
factor,  the  product  is  the  second  power.  If  the  root  is 
employed  three  times  as  a  factor,  it  is  raised  to  the  3d  power; 
if  5  times,  to  the  5th  power,  &c.  Thus,  2  is  the  1st  power  of 
2or2l.  2x  2  or  4,  is  the  2d  power  of  2,  or  22.  2x2x2 
or  8,  is  the  3d  power  of  2,  or  23.  2  X  2  X  2  X  2  X  2  or  32, 
is  the  5th  power  of  2,  or  25.  The  power  is  usually  denoted 
by  a  small  figure  over  the  right  of  the  root,  called  the  expo 
nent,  or  index.  When  there  is  no  exponent,  the  number  is 
regarded  as  the  1st  power. 

The  second  power  is  often  called  the  square,  because 
the  number  of  square  feet  in  any  square  surface,  is  obtained 
by  multiplying  the  number  of  feet  in  one  side  by  itself. 

The  third  power  is  often  called  the  cube,  because  the 
number  of  cubic  feet  in  any  cubical  block,  may  be  obtained 
by  raising  the  number  of  feet  in  one  side  to  the  3d  power. 


§84.]  INVOLUTION.  275 

The  4th  power  is  sometimes  called  the  bi-quadrate,  or  the 
square  squared ;  the  5th  power,  the  first  sursolid ;  the  6th 
power,  the  square  cubed,  or  the  cube  squared;  the  7th  power, 
the  second  sursolid;  the  8th  power,  the  bi-quadrate  squared; 
the  9th  power,  the  cube  cubed ;  the  10th  power,  the  1st 
sursolid  squared,  &c. 

If  the  exponents  of  any  two  powers  of  the  same  number 
be  added,  we  shall  obtain  the  exponent  of  their  product. 
Thus  63x65  =  6x6x6x6x6x6x6x6  =  68;42X 
43  =  4x4x4x4x4  =  4s. 

In   any  two   powers  of  the  same  number,  if  we  sub 
tract  the  smaller  exponent  from  the  larger,  we  shall  ob 
tain   the    exponent   of    their   quotient.     Thus  68  -~  65  = 
6x6x6x6x6x6x6  x  6  =  Q>,Qx6  =  & 
6x  6x6x  6x6 

We  may  represent  any  power  of  a  number  by  multiplying 
its  exponent.  Thus,  the  7th  power  of  5  is  57 ;  the  3d  power 
of  22  is  26,  because  22  X  22  X  22  =  26.  These  properties 
form  the  basis  of  the  system  of  Logarithms. 

1.  What  is  the  2d  power  of  6 ?  the  3d  power? 

2.  Find  the  value  of  .94;  123;  (£)5;  29. 

3.  Find  the  value  of  164;  1.64;  .164;  (U)3. 

4.  What  is  the  square  of  13.68  ?  of  9|  ? 

5.  What  is  the  difference  between  34  and  43  ? 

6.  What  is  the  value  of  I17;  37;  23  X  22? 

7.  What  power  of  9  is  equivalent  to  95  X  93;  92  X  910; 
94X  96;  9  x  97X  98? 

8.  Multiply  1279  by  1277,  and  divide  the  product  by 
12715. 

9.  Divide  319  by  319;  178  by  175;  427  by  426. 

10.  What  is  the  sixth  power  of 


276  INVOLUTION   AND   EVOLUTION.     [ART.  XVUI. 

11.  WTiat  is  the  9th  power  of  53  ?  the  12th  power  of  185? 
the  24th  power  of  172  ? 

85.    EVOLUTION. 

EVOLUTION  is  the  process  by  which  we  discover  the  root 
of  any  given  power.  Thus,  3  is  the  2d  root  of  9,  the  3d  root 
of  27,  the  5th  root  of  243,  because  9  =  32;  27  =  3";  243 
=  35.  So  the  2d  or  square  root  of  49  is  7  ;  the  3d  or  cube 
root  of  125  is  5  ;  the  4th  root  of  16  is  2  ;  the  5th  root  of 
1024  is  4,  &c.  We  may  denote  a  root  by  a  radical  sign, 
or  by  a  fractional  exponent. 

The  radical  sign  is  V  ,  and  when  employed  by  itself 
denotes  the  square  root.  If  we  wish  to  denote  the  3d,  5th, 
7th,  &c.  root,  the  index  of  the  root  is  written  above  the 
radical  sign,  thus,  V  ,  V  ,  &c.  In  fractional  expo 
nents,  the  numerator  expresses  the  power  of  the  number, 

and  the  denominator  expresses  the  root.  Thus,  (27)  3  = 
3/27;  (16)*=  V16"3;  (32)2=  v~324,  &c. 

The  product,  or  the  quotient,  of  two  second,  third,  or 
other  roots,  is  the  2d,  3d,  &c.,  root  of  the  product  or  quo 
tient.  Thus,  V27  X  ^125  =  V27X  125  or  ^"3375.  For 
27  =  33  =  3x  3  x  3,and  125  =  5  X  5x5.  Then27xl25 
=  3x3x3x5x5x5  =  3x5x3x5x3x5  =153. 
Therefore  3/27  X  125  =  15.  In  a  similar  manner  it  may 
be  shown  that  3/3375-7-  ^125  =  tyW. 

The  power  of  any  root  may  be  obtained  by  multiplying 

j2 

the  fractional  exponent.     Thus,  the  4th  power  of  27"3  = 
271     For  by  the  last  proposition,  ^/272  X  ^~272 


The  root  of  any  power  or  root  may  be  obtained  by  di 
viding  the  exponent  by  the  index  of  the  desired  root.     Thus, 


§85.]  EVOLUTION. 

This  is  the  converse  of  the  last  proposition.     For  if  the 

131 

3d  power  of  3'o  is  3*5',  or  3°,  the  3d  root  of  3*  must  be 
3  TV. 

If  the  numerator  and  denominator  of  fractional  indices 
be  multiplied  or  divided  by  the  same  number,  the  value  of 
the  quantity  is  not  altered.  Thus,  3g  =  3^2  =  3!.  por 
the  multiplication  of  the  numerator  involves  the  number  to 
a  certain  power,  and  the  multiplication  of  the  denominator 
extracts  the  corresponding  root.  Then  the  3d  root  of  the 
3d  power,  the  5th  root  of  the  5th  power,  £c.,  is  the  1st 
power. 

We  may  multiply  or  divide  any  two  roots  of  the  same 
number,  by  adding  or  subtracting  the  fractional  exponents. 

Thus,  ^/Tx  ^2  =  2*  +  s  =  2&;   3/5+  4/5  =  53-?  = 

_i 
51-.     For  by  the  last  proposition  we  have  4/2  X  v^  = 

22  x  ^23,  which  is  equivalent  to  V23  or  2*.     Also  ^73 


or 


When  the  exact  root  of  a  number  can  be  obtained,  it  is 
called  a  rational  number.  An  irrational  number,  or  surd, 
is  one  whose  exact  root  cannot  be  obtained.  Thus,  */!&, 
^27,  V64,  $/81,  are  rational  numbers,  equivalent  to  4,  3, 

4,  3,  respectively.     But  ^  */  19,  V  l^are  all  surds,  and 
their  roots  can  only  be  obtained  approximately. 

A  number  which  has  a  rational  root,  is  called  a  perfect 
power.  Thus,  16  is  a  perfect  2d  power,  and  a  perfect  4th 
power,  but  an  imperfect  power  of  any  other  degree.  But 

5,  7,  12,  &c.,  are  imperfect  powers  of  any  degree. 

1.  What  is  the  square  root  of  9  ?  the  cube  root  of  8  ? 

2.  What  is  the  4th  root  of  81  ?  the  5th  root  of  32  ? 

3.  What  is  the  value  of  9jT',  -5^"  ;    ^04  ;  64  ™  ? 


278  INVOLUTION   AND   EVOLUTION.     [ART.  XVIII. 

4.  What  is  the  product  of  V8  by  t/12;  of  ;/9  by  7^? 

5.  Multiply  ^3~by  tyF;  7*  by  #7;  4*  by  4/41 

6.  Divide  6*  by  6*;   4^5 by  4/5;  17*  by  ^17. 

7.  Find  the  4th  power  of  v/9;  the  6th  power  of  85. 

8.  What  is  the  cube  root  of  76;  the  5th  root  of  II7?  ? 

8G.    ROOTS  OF  ALL  POWERS.* 

When  the  exponent  of  a  power  can  be  resolved  into  two 
or  more  factors,  by  successively  extracting  the  roots  de 
noted  by  those  factors,  we  may  obtain  the  root  desired.  Thus, 
as  12  =  3  X  2  X  2,  the  cube  root  of  the  square  root  of  the 
square  root  of  a  number,  is  equal  to  the  12th  root.  So  the 
square  root  of  the  square  root  is  the  4th  root;  the  cube  root 
of  the  cube  root  is  the  9th  root;  the  cube  root  of  the  square 
root  is  the  6th  root,  &c. 

The  following  rule  will,  however,  be  generally  found  more 
convenient  for  determining  the  roots  of  all  powers  greater 
than  the  cube. 


*  It  does  not  seem  necessary  to  give  the  ordinary  rules  for  the  ex 
traction  of  the  square  and  cube  roots,  as  the  pupil  is  supposed  to  be 
already  familiar  with  them.  But  the  following  formulas  will  probably 
be  found  useful,  in  explaining  the  usual  method  of  finding  the  trial  and 
complete  divisors. 

The  square  of  any  number  consisting  of  tens  and  units  = 
The  square  of  the  tens  -f- 
(2  X  the  tens  +  the  units)  X  the  units. 
The  cube  of  any  number  consisting  of  tens  and  units  = 
The  cube  of  the  tens  -+- 
3  X  the  square  of  the  tens  +      \ 
3  X  the  tens  X  the  units  +         r  X  the  units. 

the  square  of  the  units. 
The  entire  portion  of  the  root  which  has  been  found,  at  any  step, 
may  be  considered  as  the  tens,  and  the  next  root  figure  will  then  rep 
resent  the  units. 


§86.]  ROOTS    OF   ALL   POWERS.  279 


GENERAL  RULE. 

At  the  left  of  the  number  whose  root  is  required,  arrange 
as  many  columns  as  are  equal  to  the  index  of  the  root, 
writing  1  at  the  head  of  the  first  or  left  hand  column,  and 
zero  at  the  head  of  each  of  the  others. 

Divide  the  number  into  periods  of  as  many  figures  as  the 
index  of  the  root  requires.  Write  the  root  of  the  left  hand 
period  as  the  first  figure  of  the  true  root. 

Multiply  the  number  in  the  first  column  by  the  root  fig 
ure,  and  add  the  product  to  the  second  column;  add  the 
product  of  this  sum  by  the  root  figure  to  the  third  column, 
and  so  proceed,  subtracting  the  product  of  the  last  column 
from  the  given  number. 

Repeat  this  process,  stopping  at  the  last  column,  and  thus 
proceed,  stopping  one  column  sooner  each  time,  until  the 
last  sum  falls  in  the  second  column. 

To  determine  the  second  root  figure,  consider  the  number 
in  the  last  column  as  a  trial  divisor,  and  proceed  with 
the  second  root  figure  thus  obtained/  precisely  as  with  the 
first. 

Continue  the  operation  until  the  root  is  completed,  or  the 
approximation  carried  as  far  as  is  desired. 

In  order  to  avoid  error,  observe  carefully  the  value  of 
each  root  figure  and  each  product.  Thus,  if  the  first  root 
figure  is  hundreds,  the  number  in  the  second  column  will  be 
hundreds, — in  the  third,  ten  thousands, — in  the  fourth, 
millions,  &c. 

EXAMPLES  FOR  ILLUSTRATION. 
1.  What  is  the  third  root  of  205692449327  ? 

a  If  the  root  figure  thus  found  proves  too  large,  erase  it  and  try  a 
smaller  number. 


280 


INVOLUTION   AND   EVOLUTION.      [ART.  XVIII. 


0 

5  thous. 

0 

25  mill. 

205692449327(5903 
125  bill. 

5 
5 

25  c.  d. 

50 

806,92 
80379  mill. 

10 
5 

159  him. 
9 

168 
9 

(1)  75  t.  d. 
1431  ten  thous. 

8931  c.  d. 
1512 

3134,49327 
313449327  un. 

(2)  10443  t.  d. 
53109  un. 

(1) 


(2)   17703  un. 


104483109  c.  d. 


The  complete  divisors  are  marked  c.  d.,  the  trial  divisors,  t.  d. 
The  figures  at  which  the  new  additions  commence  are  marked 
(1),  (2).  The  partial  dividends  by  which  each  root  figure  is  de 
termined,  are  distinguished  by  a  comma.  They  always  terminate 
with  the  first  figure  of  the  period  that  is  annexed.  The  abbre 
viations,  thous.,  mill.,  &c.,  show  the  value  of  the  figures  against 
which  they  are  placed. 

2.  Extract  the  5th  root  of  858533232.56832. 


L        0 
6  tens. 

6 

6 

12 

6 

000                  858533232.56832(61.2 
36  hund.           216  thous.         1296  ten  thou  .  7776  hunt!  .  thous. 

~36~                    216"                   1296  c.  d.          8093,3232 
72                     648                    5184                    66996301 

108                    "ieT              (l)6480t.  d.             139369315,6832 
108                    1296                       2196301  un.       1393693156832 

18 
6 

216 
144 

(1)2160                     66996301  c.d. 
36301  un.           2232904 

thous. 

24 
6 

(1)  301  un. 

(1)  360 
301  un 

2196301          (2)  69229205  t.  d. 
36603                   4554528416  ten 

36301 
302 

2232904                6968465784  16  c  .  d  . 
36906 

302 
1 

36603 
303 

(2)  2269810 
7454208  thous. 

303 
1 

~304 
1 

36906 
304 

2277264208 
hund. 

(2)  37210 
6104 

(2)  3052  tenths  3727104 

The  additions  to  the  left  hand  column  may  be  made  mentally, 
and  thus  shorten  the  labor.  There  are  other  abbreviations,  for 
which  the  student  is  referred  to  the  Chapter  on  Approximations. 


§86.] 


ROOTS   AND   POWERS. 


281 


•g 


S     2°     2 

—      =r  !   =r 

nil 


II 


S 

Sin 


I 

i! 


|g|£ 

|g|£|5  S|8|S|§|8|810 


a     M 


S   i     O    !     00    I     S    |     CO    !     O   i     00 

B  !  I  i  g  !  8  i  8  i  §  ;  S 


* 


OD         O 
ill 


8  1  8 


S2    ^ 


td 


O 

c 

H 

02 


O 

^ 

SI 

w 

CQ 


282  INVOLUTION   AND    EVOLUTION.     [ART.  XVIH. 

The  first  root  figure  in  each  of  the  following  examples  may  be 
found  by  the  table  of  Powers  and  Boots. 

1.  Extract  the  square  root  of  350026681. 

2.  Extract  the  square  root  of  3 ;  5 ;  6.5. 

3.  Extract  the  cube  root  of  2924207. 

4.  Extract  the  cube  root  of  13;  12.5. 

5.  Extract  the  fifth  root  of  65.7748550151. 

6.  Extract  the  7th  root  of  1.246688292353624506368. 

87.    APPLICATION  OF  THE  SQUARE  ROOT. 

The  areas  of  any  similar  figures  are  proportioned  to  the 
squares  of  their  like  dimensions. 

The  area  of  any  circle  is  equal  to  the  square  of  its  di 
ameter  multiplied  by  .7854. 

The  circumference  of  a  circle  is  equal  to  its  diameter 
multiplied  by  3.1416.a 

The  area  of  a  triangle  is  equal  to  the  base  multiplied  by 
half  the  height. 

In  any  right-angled  triangle,  the  square  of  the  longest 
side  is  equal  to  the  sum  of  the  squares  of  the  other  two 
sides. 

The  distance  through  which  bodies  fall,  when  falling 
freely,  are  as  the  squares  of  the  times.  In  a  vacuum,  a 
body  would  fall  16yLft.  in  1  second.  Then  we  have  the 
proportion,  letting  n  represent  any  number  of  seconds, 

sec.       sec.     ft.         ft. 

(I)2  :  n2  : :  16^  :  distance  in  n  seconds. 

Any  three  terms  of  this  proportion  being  given,  the 
fourth  may  be  readily  founi.  But  it  should  be  remarked, 
that  in  consequence  of  the  resistance  of  the  air,  the  space 

»  The  more  exact  ratio  is,  3.14159265358979323846264338328. 


§87.]  APPLICATION    OP   THE    SQUARE   ROOT.  283 

actually  fallen  through  is  somewhat  less  than  that  given  by 
the  formula. 

If  b  represents  the  base  of  a  right-angled  triangle,  p 
the  perpendicular,  and  li  the  hypothenuse,  li  =  \/p2H-  &2; 
I  =  V  li1—  p2',  p  =  ^h2  —  b2. 

The  square  root  of  the  area  of  any  surface  will  give  the 
side  of  a  square,  equal  in  area  to  the  given  surface. 

EXAMPLES. 

1.  What  is  the  diameter  of  a  circle  that  is  16  times  as 
large  as  one  whose  diameter  is  13  feet  ?  Ans.  52ft. 

2.  The  area  of  a  circle  is  7632  feet;  what  is  the  diame 
ter?  Ans.  98.5ft. 

3.  A  horse  is  fastened  to  a  post  in  the  centre  of  a  field. 
What  is  the  length  of  a  rope  that  will  allow  him  to  graze 
an  acre?  Ans.  7.136  rods. 

4.  A  ladder  75  feet  long,  rests  against  the  trunk  of  a  tree 
at  a  point  50  feet  from  the  ground.    How  far  is  the  foot  of 
the  ladder  from  the  root  of  the  tree  ?  Ans.  55.9ft. 

5.  The  length  of  a  room  is  18  feet,  and  the  width  12 
feet.     What  is  the  distance  between  the  opposite  corners  ? 
What  length  of  rope  would  reach  from  an  upper  corner  to 
the  opposite  lower  corner,  the  height  being  10  feet  ? 

Ans.  21.6ft.;  23.832ft. 

6.  The  circumference  of  a  circle  is  29  rods.     What  is 
the  side  of  a  square  having  an  equal  area  ? 

Ans.  8.18  rods. 

7.  Two  ships  left  the  same  port;  one  sailed  125  miles 
north,  the  other  100  miles  east.     How  far  were  they  then 
apart  ?  Ans.  160m.  24.96r. 

8.  A  kite  accidentally  lodged  in  the  top  of  a  tree,  but 
the  line  breaking,  I  measure  its  length,  which  is  210  feet. 


284  INVOLUTION   AND   EVOLUTION.    [ART.  XVIII. 

What  is  the  height  of  the  tree,  the  foot  being  189  feet 
from  my  standing  place  ?  Ans.  91.53ft. 

9.  Desiring  to  know  the  height  of  a  precipice,  I  drop  a 
stone  from  the  summit,  and  observe  by  my  watch  that  it 
strikes  the  ground  in  3£  seconds.     What  is  the  height  ?a 

Ans.  197.02ft, 

10.  A  bag  of  sand  is  dropped  from  a  balloon  11  miles 
above  the  surface  of  the  earth.     How  long  will  it  be  in 
falling?1  Ans.  20.25sec. 

When  one  number  bears  the  same  ratio  to  a  second  as 
the  second  does  to  a  third,  the  second  number  is  called  a 
mean  proportional  between  the  other  two.  Thus,  in  the 
proportion  3  :  6  :  :  6  :  12,  6  is  a  mean  proportional  between 
3  and  12. 

The  mean  proportional  between  any  two  numbers  is  equal 
to  the  square  root  of  their  product. 

11.  Find  a  mean  proportional  between  7  and  252. 

12.  Find  a  mean  proportional  between  .75  and  12. 

13.  Find  a  mean  proportional  between  *  and  ? J5-. 

14.  Find  a  mean  proportional  between  ^J^  and  .875. 

15.  Find  mean  proportionals  between  ^  and  16 ;  5  and 
6;  25  and  13;  §-  and  f. 

88.    APPLICATION  OP  THE  CUBE  ROOT. 

The  solid  contents  and  the  weights  of  similar  bodies  are 
to  each  other  as  the  cubes  of  their  diameters,  or  of  their 
similar  sides. 

The  solid  contents  of  a  sphere  may  be  found  by  multi 
plying  the  cube  of  the  diameter  by  .5236. 

The  cube  root  of  the  solid  contents  of  any  body,  will  give 
the  side  of  a  cube,  equal  in  solidity  to  the  given  body. 

a  No  allowance  is  made  for  resistance  of  the  air,  in  the  answers  that 
are  given. 


§88.]  APPLICATION    OF   THE   CUBE   ROOT.  285 

EXAMPLES. 

1.  What  are  the  solid  contents  of  the  earth,  supposing 
it  a  perfect  sphere,  whose  diameter  is  7920  miles  ? 

Ans.  260120860876.8  cubic  miles. 

2.  If  a  ball  2  inches  in  diameter,  weighs  1J  pounds, 
what  would  be  the  weight  of  a  similar  ball  6  inches  in 
diameter?  Ans.  40 Jib. 

3.  What  is  the  side  of  a  cubical  box  that  will  hold  1 
bushel?  Ans.  12.907in. 

4.  What  is  the  side  of  a  cubical  pile  that  contains  256 
cords  of  wood  ?  Ans.  32ft. 

5.  If  a  tree  1  foot  in  diameter,  yields  2  cords  of  wood, 
how  much  wood  is  there  in  a  similar  tree  that  is  3ft.  6in.  in 
diameter  ?  Ans.  85f  cords. 

6.  If  a  pound  avoirdupois  of  gold  is  worth  $200,  and  a 
cubic  inch  weighs  11£  oz.,  what  would  be  the  value  of  a 
gold  ball  1  foot  in  diameter  ?  Ans.  $243000. 

7.  What  is  the  diameter  of  a  ball  that  weighs  27  times 
as  much  as  one  3  feet  6  inches  in  diameter  ? 

Ans.  10ft.  Gin. 

8.  If  a  hollow  sphere  3  feet  in  diameter  and  2J  inches 
thick,  weighs  12  tons,  what  would  be  the  dimensions  of  a 
similar  sphere  that  would  weigh  324  tons  ? 

Ans.  Diameter  Oft. ;  thickness  7  inches. 

9.  What  is  the  side  of  a  cubical  block  of  wood,  that  weighs 
as  much  as  a  sphere  of  the  same  material,  15  inches  in 
diameter?  Ans.  12.09  inches. 

10.  The  length  of  a  ship's  keel  is  70ft.,  the  breadth  of 
beam  25ft.,  and  the  depth  of  the  hold  12Jft.     Required 
the  dimensions  of  another  vessel,  built  on  the  same  model, 
but  of  twice  the  tonnage. 


286  PROGRESSION,    OR   SERIES.  [ART.  XIX. 

11.  If  a  ship  whose  keel  measures  90  feet,  carries  420 
tons,  what  will  be  the  tonnage  of  a  similar  vessel  with  a 
keel  60ft.  long? 


XIX.    PROGRESSION,  OR  SERIES. 
89.     ARITHMETICAL  AND  GEOMETRICAL  PROGRESSION. 

LET  a  represent  the  less  extreme  of  a  series,  I  the 
greater  extreme,  n  the  number  of  terms,  s  the  sum  of  all 
the  terms  in  an  arithmetical  series,  p  the  product  of  all  the 
terms  in  a  geometrical  series,  d  the  arithmetical  difference, 
and  r  the  geometrical  ratio.  Then 

In  Arithmetical  Progression, 


rion. 

In  Geometrical 

Progression. 

(i) 

l=aX  rn~l 

(5) 

(2) 

p  =  V  (a  X  I) 

(6) 

I 

(3) 

a  =  

rn-l 

(7) 

(4) 

r..Jl 

N  a 

(8) 

a  =  l  -  (n  -  1)  d 


n  —  1 


In  comparing  these  tables,  we  see  that 

addition  corresponds  to  multiplication; 
subtraction       "  "  division; 

multiplication1'  "  involution; 

division  "  "  evolution. 

If,  therefore,  we  had  a  series  of  numbers  bearing  the 
same  ratio  to  the  natural  series,  as  an  Arithmetical  to  a 
Geometrical  Progression,  the  labor  of  multiplication  would 
be  reduced  to  that  of  simple  addition,  and  involution  to 
simple  multiplication.  Such  a  series  constitutes  a  TABLE 
OP  LOGARITHMS. 


§89.]  ARITHMETICAL  PROGRESSION.  287 

EXAMPLES. 

1.  Determine  the  value  of  n,  when  a,  d}  and  I  are  given, 
Iby  the  2d  method  of  analysis,  stated  in  Section  57. 

By  formula  (1)  we  perceive  that  if  1  be  subtracted  from  n,  the 
remainder  multiplied  by  d,  and  a  added  to  the  product,  the  sum 
will  be  I.  Reversing  the  operation,  if  we  subtract  a  from  I,  divide 
the  remainder  by  d,  and  add  1  to  the  quotient,  the  sum  will  be  n. 

d 

2.  From  formula  (2)  find  the  value  of  I,  when  a,  n,  and 
s  are  given,  and  the  value  of  a,  when  I,  n}  and  s  are  given. 

Ans.  *=*-«;  a  =  * -I 

n  n 

3.  Determine  the  value  of  n,  when  the  values  of  a,  I,  and 
s  are  known.  2s 


4.  From  formulas  (1)  and  (2)  determine  the  value  of  a, 
when  dj  n,  and  s  are  known. 

Substituting  for  I  in  formula  (2),  its  value  in  formula  (1),  we 


Reversing  all  the  operations  that  must  be  performed  on  a  to 

produce  this  result,  we  find  that  a  =  (_!  —  (n  —  1)  d}  -j-  2. 

n  ' 

5.  From  formulas  (2)  and  (3),  how  may  we  find  the  value 
of  lj  when  d,  n,  and  s  are  known  ?  a 


Ans.  l=*—+n^-ld+ 


a  There  are  two  formulas,  which  the  pupil  could  hardly  be  expected 
ro  obtain,  without  considerable  knowledge  of  Algebra.  They  are, 
therefore,  inserted  here,  in  order  that  he  may  have  all  the  formulas 
that  are  necessary  to  solve  any  question  in  Arithmetical  Progression 
that  can  possibly  occur. 

When  a,  d,  and  s  are  given, 


_ 


288  PROGRESSION,  OR    SERIES.  [ART.  XIX. 

6.  A  laborer  agreed  to  dig  a  well  39  yards  deep,  for 
which  he  was  to  be  paid  as  follows  :  75  cents  for  the  first 
yard,  31.25  for  the  second  yard,  and  so  on,  increasing  50 
cents  for  each  subsequent  yard.     What  would  the  last  yard 
cost,  and  what  would  he  receive  for  the  whole  job  ? 

2d  Ans.  $399.75. 

7.  The  formula  for  determining  the  sum  of  any  geomet 
rical  series  is  s  =    ^^~a-     Determine  from  this  formula, 
the  value  of  r  when  s,  a,  and  I  are  given. 

Ans.  r= 

s  —  I 

8.  When  one  of  the  extremes,  the  ratio,  and  the  sum  of 
the  terms  are  given,  how  would  you  find  the  other  extreme  ? 
Give  separate  answers  for  each  extreme. 

Ans.  a  =  lX  r — (r  —  1)  X  s. 

I  =  i • 

r 

9.  What  is  the  sum  of  the  series  2,  1,  J,  i,  &c.,  to  infi 
nity  ?    (The  last  term  in  any  infinite  decreasing  series  is  0.) 

Ans.  4. 

10.  If  a  man  commences  at  21  years  of  age,  and  annu 
ally  puts  $500  at  compound  interest,  how  much  will  he  be 
worth  when  he  is  50  years  old  ?  Ans.  836819.90. 

11.  Insert  2  mean  proportionals  between  1  and  343.    (As 


When  d,  Z,  and  s  are  given, 


___ 
—  Sds 


The  sign  following  d  in  the  second  formula,  is  sometimes  +,  and 
sometimes  —  .  The  proper  sign  can  easily  be  determined  by  trial. 

a  If  (Zx  r  —  a)  ~-(r  —  !)=«,  r  X  *  —  «=^  X  r  —  a.  Then 
r  x  s—-lxr  +  s  —  a-  Subtracting  I  X  r  from  r  X  s,  we  have  r  X  * 
—  l^r^s  —  a.  ButrX*  —  rX  Z  —  ^X  (*  —  Z);  and  therefore, 
dividing  by  a  —  Z,  we  obtain  the  answer,  r  =  (s  —  o)  -r-  (s  —  Z).  This 
analysis  will  be  more  difficult  to  follow  than  any  of  those  required  in 
arithmetical  progression  ;  but  the  pupil  should  pass  nothing  over  until  he 
understands  it  perfectly. 


§90.]  IIARMONICAL   PROGRESSION.  289 

there  are  to  be  2  means,  the  number  of  terms  is  4,  and  the 
extremes  1  and  343.)  Ans.  7,  49. 

12.  Insert  5  mean  proportionals  between  4  and  2916. 

Ans.  12,  36,  108,  324,  972. 

13.  Every  oviparous  fish  deposits  annually,  at  the  spawn 
ing  season,  many  thousands  of  ova.     If  we  estimate  the 
average  number  deposited  by  each  pair  of  herrings  to  be 
only  2000,  to  what  number  would  the  offspring  of  a  single 
pair  amount  in  the  eighth  year,  supposing  that  every  egg 
produced  a  fish  ? 

Ans.   2  septillion,  a  number  which  would  make  a 
mass  larger  than  the  whole  globe. 

14.  According  to  some  experiments,  it  has  been  found 
that  one  stem  of  the  hyoscyamus  sometimes  produces  more 
than  50000  seeds.    At  this  rate,  if  every  seed  should  produce 
a  fertile  plant,  what  number  of  plants  would  be  contained  in 
the  fourth  crop  from  a  single  seed  ? 

Ans.  6250  quadrillion,  a  number  that  the  whole  surface 
of  the  earth  would  not  be  sufficient  to  contain. 

15.  If  the  human  race,  after  making  a  proper  deduction 
for  those  who  died,  had  doubled  every  twenty  years,  how  many 
of  the  descendants  of  Adam  would  have  been  living  when  he 
was  500  years  old  ?  Ans.  33554430. 

9O.    HARMONIOAL  PROGRESSION.* 

When  three  numbers  are  such  that  the  first  is  to  the 
third,  as  the  difference  between  the  first  and  second  is  to  the 
difference  between  the  second  and  third,  they  are  said  to  be 
in  HARMONICAL  PROPORTION  ;  and  a  series  of  numbers,  in 
continued  harmonical  proportion,  constitutes  a  HARMONI 
CAL  PROGRESSION. 

The  reciprocal  of  a  number,  is  the  quotient  of  1  by  the 

a  So  called,  because  if  a  musical  string  be  divided  in  harmonical 
proportion,  the  different  parts  will  vibrate  in  unison. 

19 


290  PROGRESSION,   OR   SERIES.  [ART.  XIX 

number.  Thus,  ±  is  the  reciprocal  of  2 ;  4  is  the  reciprocal 
of  -j;  |  is  the  reciprocal  of  f,  &c.  The  reciprocals  of  any 
equidifferent  series  form  a  liar monical  proportion. 

I.  Two  numbers  being  given,  to  find  a  third  in  harmoni- 
cal  proportion. 

Consider  the  reciprocals  of  the  numbers  as  two  terms  of 
an  equidifferent  series.  The  third  term  will  be  the  recipro 
cal  of  the  number  sought. 

Find  a  third  harmonical  proportional  to  120  and  40. 

The  reciprocals  are  TJu»  and  4^,  or  T3(j-  The  third  term  of 
the  equidifferent  series  is  yf  Q,  and  its  reciprocal  24  is  the  har 
monical  proportional  sought. 

II.  To  insert  any  number  of  harmonical  means  between 
two  numbers. 

Find  as  many  arithmetical  means  between  the  reciprocals 
of  the  given  numbers.  These  means  will  be  the  reciprocals 
of  the  harmonical  means. 

Insert  4  harmonical  means  between  20  and  120. 

The  reciprocals  are  ^4  and  y^o,  or  y|o  and  y^jy.  The  four 
arithmetical  means  are  yf  Q>  y|(TJ  llo?  anc^  T!O>  whose  recipro 
cals  are  24,  30,  40,  and  60, — the  desired  harmonical  means. 

EXAMPLES. 

1.  The  first  two  terms  of  a  harmonical  progression  are 
60  and  30.     Required  the  ten  succeeding  terms. 

2.  The  first  two  terms  of  a  harmonical  proportion  are 
348075  and  69615.     Find  the  six  succeeding  terms. 

3.  Insert  6  harmonical  means  between  630  and  5040 

4.  Insert  8  harmonical  means  between  10  and  60. 

5.  Insert  2  harmonical  means  between  J  and  J. 

6.  Insert  4  harmonical  means  between  J  and  T\j. 


§91.]  COMPOUND   INTEREST.  291 

Ol,     COMPOUND  INTEREST. 

Compound  Interest  may  be  computed  by  Geometrical 
Progression ;  a  =  the  amount  of  $1  for  the  time  that  should 
elapse  between  two  successive  payments  of  interest ;  r  =  a ; 
n  =  the  number  of  payments. 

The  labor  of  computing  Compound  Interest,  may  be  abridged 
by  a  table  in  which  the  amount  of  $1  is  computed  at  different 
rates,  and  for  a  number  of  years.  (See  Table  I.,  p.  293.) 

I.  To  find  the  amount  of  any  sum  by  the  Table,  multiply  the 
given  sum  by  the  amount  of  $1  for  the  given  time. 

EXAMPLE.  What  will  be  the  amount,  at  7  per  cent,  compound 
interest,  of  $200  for  15yr.? 

$1  in  15yr.  at  7  per  cent,  amounts  to  2.759031,  and  2.759031  X 
$200  =  $551. 8062. 

II.  To  compute  compound  discount,  or  to  find  the  present  worth 
at  compound  interest,  of  any  sum  due  at  a  future  time,  divide  the 
given  sum  by  the  amount  of  $1  for  the  given  time. 

EXAMPLE. — When  money  is  worth  5  per  cent,  compound  interest, 
what  is  the  present  worth  of  $5000  due  in  19yr.  4mo.  24dy.  ? 

$1  at  5  per  cent,  would  amount  in  19yr.  4mo.  24dy.  to  $2.577489, 
and  $5000  -+.  2.577489  =  $1939.87. 

III.  To  find  the  time  in  which  any  principal  will  amount  to  a 
given  sum,  divide  the  amount  by  the  principal,  and  look  for  the  quotient 
in  the  Table,  under  the  given  rate. 

EXAMPLE. — In  what  time,  at  6  per  cent,  compound  interest,  will 
$25  amount  to  $48  ? 

|f  =  1.92.  $1  would  amount  to  1.898299  in  11  years,  or  to 
2.012196  in  12yr. 

1.92  exceeds  1.898299  by  .021701,  and  2.012196  exceeds 
1.898299  by  .113897.  Then  if  the  gain  in  12  months  is  .113897, 
in  what  time  would  there  be  a  gain  of  .021701  ? 

.113897  :  .021701  :  :  12mo.  :  2mo.  8dy.  very  nearly. 

IV.  To  find  the  rate  at  which  any  principal  will  amount  to  a 
given  sum  in  a  given  time,  divide  the  amount  by  the  principal,  and 
look  for  the  quotient  in  the  Table,  opposite  the  given  time. 

EXAMPLE. — At  what  rate  of  compound  interest,  will  $250  amount 
to  $550  in  18  years? 


292  PROGRESSION,    OR   SERIES.  [ART.  XIX. 


Is?  —  2-2-  ^  tlie  line  of  18  years>  we  find  2'2  under  4£  Per 
cent. 

EXAMPLES. 

1.  Find  the  amount  of  $637.25,  at  5  per  cent,  compound 
interest,  for  16yr.  3mo.  15dy.  An*.  81411.32. 

2.  Allowing  7  per  cent,  compound  interest,  what  is  the 
present  worth  of  $1000,  due  in  35yr.  5mo.  6dy.? 

Am.  $90.91. 

3.  At  6  per  cent,  compound  interest,  in  what  time  will 
$250  amount  to  $1000  ?  Ans.  23yr.  9mo.  13dy. 

4.  At  what   rate    of  compound   interest  will   $127.75 
amount  to  $201.22,  in  lOyr.  3mo.  24dy.  ? 

Ans.  4£  per  cent. 

92.    ANNUITIES. 

Any  sum  of  money  to  be  paid  regularly,  at  stated 
periods,  is  called  an  ANNUITY.  The  payment  may  be 
stipulated  for  a  given  number  of  years,  in  which  case  it  is 
called  an  annuity  certain,  or  it  may  be  dependent  upon 
some  particular  circumstance,  as  the  life  of  one  or  more 
individuals.  The  latter  is  called  a  contingent  annuity.  A 
perpetual  annuity,  is  one  which  can  only  be  terminated  by 
the  grantor,  on  the  payment  of  a  sum  whose  interest  will 
be  equivalent  to  the  annuity.  Of  this  character  is  the 
consolidated  debt  of  England. 

An  annuity  in  possession,  is  one  on  which  there  is  a  pres 
ent  claim  '}  an  annuity  in  reversion,  or  deferred  annuity,  is 
one  that  does  not  commence  until  the  lapse  of  a  stated 
time,  or  the  occurrence  of  some  uncertain  event,  as  the 
death  of  an  individual. 

The  present  worth  of  an  annuity,  is  the  sum  which,  at 
compound  interest  for  the  time  of  its  duration,  would 
amount  to  the  sum  of  all  the  payments,  each  being  placed 
at  compound  interest  as  it  became  due. 


§92.] 


ANNUITIES. 
TABLE   I. 


293 


SHOWING   THE   AMOUNT   OP   $1.00,   AT   COMPOUND   INTEREST,   FROM 
1  YEAR  TO  50. 


Year. 

3  p.  cent. 

3|  p.  cent. 

4  p.  cent. 

4£  p.  cent. 

5  p.  cent.  6  p.  cent. 

7  p.  cent. 

1 
2 
3 
4 
5 

1.030000 
1.060900 
1.092727 
1.125509 
1.159274 

1.035000 
1.071225 
1.108718 
1.147523 

1.187686 

1.040000 
1.081600 
1.124864 
1.169859 
1.216653 

1.045000 
1.092025 
1.141166 
1.192519 
1.246182 

1.050000 
1.102500 
1.157625 
1.215506 
1.276282 

1.060000 
1.123600 
1.191016 
1.262477 
1.338226 

1.070000 
1.144900 
1.225043 
I.:-.  10796 
3.  -102552 

G 

7 
8 
9 
10 

1.194052 

1.221)874 
1.2156770 
1.304773 
1.343916 

1.229255 
1.272279 
1.316809 
1.362897 
1.410599 

1.265319 
1.315932 
1.368569 
1.423312 
1.480244 

1.302260 
1.360862 
1.422101 
1.486095 
1.552969 

1.340096 
1.407100 
1.477455 
1.551328 

1.628895 

1.418519 
1.503630 
1.593848 
1.689479 
1.790848 

1.500730 
1.605781 
1.71el86 
1.838459 
1.907151 

11 
12 
13 
14 
15 

1.384234 
1.425761 
1.468534 
1.5125LiO 
1.557967 

1.459970 
1.511069 
1.563956 
1.618694 
1.675349 

1.539454 
1.601032 
J.  665073 
1.731676 
1.800943 

1.872981 
1.947900 
2.025816 
2.106849 
2.191123 

1.G22853 
1.695881 
1.772196 
1.851945 
1.935282 

1.710339 
1.795856 
1.885649 
1.979932 

2.078928 

1.898299 
2.012196 
2.132928 
2.260904 
2.396558 

2.104852 
2.252192 
2.40:)845 
2.578534 
2.759031 

16 
17 
18 
19 
20 

1.604706 
1.652848 
1.702433 
1.753506 
1.806111 

1.733986 
1.794675 

1.857489 
1.922501 

1.989789 

2.022370 
2.113377 
2.208479 
2.307860 
2.411714 

2.182875 
2.292018 
2.406619 
2.526950 
2.653298 

2540352 
2.692773 
2.854339 
3.025599 
3.207135 

2.952164  1 
3.158815 
3.37H931 
3.61652(3 

3.86iiCfe3  | 

21 
22 
23 
24 
25 

2G 
27 
28 
29 
30 

1.860295 
1.916103 
1.973586 
2.032794 
2.093778 

2.059431 
2.131512 
2.20till4 
2.283328 
2.363245 

2.278768 
2.369919 
2.464715 
2.5D3304 
2.6t>5836 

2.520241 
2.633652 
2.752166 
2.876014 
3.005434 

2.785963 
2.925261 
3.071524 
3.225100 
3.386355 

3.399564 
3.603537 
3.819750 
4.048935 
4.291871 

4.14051-1  ! 
4.4:iOiOO 
4.740528 
5.0723(55 
5.427431  ; 

2.156591 
2.221289 

2.287928 
2.356565 
2.427262 

2.445959 
2.531567 
2.620177 

2.711878 
2.806794 

2.772470 
2.883369 
2.998703 
3.118651 
3.243397 

3.140679 
3.282009 
3.421)700 
3.584036 
3.745318 

3.555673 
3.733456 
3.920129 
4.116136 
4.321942 

4.549383 
4.822346 
5.111687 
5.418388 
5.743491 

5.807351 
6.2138(16 
6.64F836 
7.11-1255 

7.012253 

31 
32 
33 
34 
35 

2.500080 
2.575083 
2.652335 
2.731905 

2.813862 

2.905031 
3.006708 
3.111942 
3.220860 
3.333590 

3.373133 
3.50r059 
3.648381 
3.794316 
3.946089 

3.913857 
4.089981 
4.274030 
4.466361 
4.667348 

4.538039 
4.764941 
5.003188 
5.253348 
5.516015 

6.088101 
6.453387 
6.840590 
7.251025 

7.686087 

8.145110 

8.715268 
9.325337 
9.978110 
10.676578 

36 
37 
38 
39 
40 

2.898278 
2.985227 
3.074783 
3.167027 
3.262038 

3.359899 
3.460696 
3.564517 
3.671452 
3.781590 

3.450266 
3.571025 
3.696011 
3.825372 
3.959260 

4.103932 
4.2680SJO 
4.438813 
4.616366 
4.801021 

4.877378 
5.096860 
5.326219 
5.565899 
5.816364 

5.791816 
6.081407 
6.385477 
6.704751 

7.039989 

7.391988 
7.761587 
8.149667 
8.557150 

8.985008 

8.147252 
8.636087 
9.154252 
9.703507 
10.285718 

11.423939 
12.2231)14 
13.079277 
13.994b27 
14.974465 

41 
42 
43 
44 
45 

4.097834 
4.241258 
4.389702 
4.543342 
4.702358 

4.993061 
5.192784 
5.400495 
5.616515 
5.841176 

6.078101 
6.351615 
6.637438 
6.936123 

7.248248 

10.902861 
11.557033 
12.250455 

12.985482 
13.764611 

16.022677 
17.144265 
18.341363 
19.628469 
21.002461 

46 
47 

48 
49 
50 

3.895044 
4.011895 
4.132-252 
4.256219 
4.383906 

4.866941 
5.037284 
5.213589 
5.:W60(i5 

5.584927 

6.074823 
6.317816 
6.570528 
6.833349 
7.106683 

7.574420 

7.915268 
8.2714->5 
8.643IS71 
9.032636 

9.434258 
9.905971 
10.401267 
10.921333 
11.467400 

14.590487 
15.465917 
16.393872 
17.377504 
18.420154 

22.472634 
24.045718 
25.728918  ! 
27.529943  1 
29.457039  J 

294  PROGRESSION,    OR   SERIES.  [ART.  XIX. 

TABLE  II. 
THE  AMOUNT  OF  AN  ANNUITY  OF  $1.00,  FROM  1  YEAR  TO  50. 


Year. 

3  p.  cent. 

3|  p.  cent. 

4  p.  cent. 

4|  p.  cent. 

5  p.  cent. 

5£  p.  cent. 

6  p.  cent. 

2 
3 
4 
5 

1.000000 
2.030000 
3.0J0900 
4.183627 
5.309136 

1.000000 
2.035000 

3.106-22.1) 

4.214!  >43 
5.36:2466 

LOOOOOO 
2.040000 
3.121600 
4.246464 
5.416322 

1.000000 
2.045000 
3.137025 
4.278191 
5.470710 

1.000000 
2.050000 
3.152500 
4.310125 
5.525631 

1.000000 
2.055000 
3.168025 
4.342266 
5.581091 

6.888051 
8.266894 
9.721573 
11.256259- 
12.875354 

1.000000 
2.060000 
3.1KMOO 
4.374616 
5.637093 

6 
7 
8 
9 
10 

6.468410 
7.662462 
8.892336 
10.159106 
11.463879 

6.550152 
7.775*408 
9.05  IGn? 
10.3684116 
11.731393 

6.632975 
7.898294 
9.214226 
10.582795 
12.006107 

6.716892 
8.011)152 
9.380014 
10.802114 
12.2ti8210 

6.801913 
8.142008 
9.549109 
11.026564 
12.577893 

14.206787 
15.917127 
17.712983 
19.598632 
21.578564 

6.975319 
8.393838 
9.897408 
11.491316 
13.180795 

14.971643 
16.869941 
18.882138 
21.015060 
23.275971 

11 
12 
13 
14 
15 

16 
17 

18 
19 
20 

12.807796 
14.192029 
15.617790 
17.086324 
18.598914 

13.141992 
14.601902 
16.113030 
17.676986 
19.295681 

13.486351 
15.025805 
16.626838 
18.291911 
20.023588 

21.824531 
23.697512 
25.645413 
27.671229 

29.77807e 

13.841179 
15.464032 
17.  1591)13 
18.932109 
20.784054 

14.583498 
16.385590 
18.286798 
20.292572 
22.408663 

20.156881 
21.761588 
23.414436 
25.116868 
26.870374 

20971030 
22.705016 
24.499691 
26.357180 
28.279682 

22.719337 
24.741707 
26.855084 
29.063562 
31.3714^ 

23.657492 
25.840366 
28.132385 
30.539004 
33.065954 

24.641139 
26.996402 
29.481205 
32.102071 
34.868318 

25.672528, 
28.212880 
30.905053 
33.759992 
36.785592: 

21 
22 
23 
24 
25 

28.676486 
30.536780 
32.452884 
34.426470 
36.459264 

30.269471 
32.328902 
34.460414 
36.666528 
38.949857 

31.969202 
34.247970 
36.617888 
39.082604 
41.645908 

33.783137 
36.303378 
38.937030 

41.689196 
44.565210 

35.719252 
38.505214 
41.430475 

44.501999 
47.727099 

37.780075  39.9027271 
40.864309  43.39229') 
44.111846  40.99582S1 
47.5379981  50.815577 
51.152588  54.fc64512 

i  26 
27 
28 
29 
30 

38.553042 
40.709634 
42.930923 
45.218850 
47.575416 

41.313102 
43.759060 
46.290627 
48.910799 
51.622677 

44.311745 
47.084214 
49.967583 
52.966286 
56.08493e 

47.570645 
50.711324 
53.993333 
57.423033 
61.007070 

51.113454 

54.669120 
58.402583 
62.322712 
66.438847 

54.905979  59.156383 
58.989109  63.705706 
63.233510  C8.52S1J? 
67.7113531  73.639798 
72.435478j  79.058186 

31 

32 
33 
34 
35 

36 
37 
38 
i  39 

40 

50.002678 
52.502759 
55.077841 
57.730177 
60.402082 

54.429471 
57.334502 
60.341210 
63.453152 
66.674013 

59.328335 
62.701469 
66.201)527 
69.857909 
73.652225 

64.752388 
68.666245 
72.756226 
77.030256 

81.496618 

70.760700 
75.29882'. 
80.063770 
85.066959 
90.320307 

77.419429.  84.80167- 
62.677498J  80.889778 
88.224700  97.343165' 
94.077122  104.L-3755 
100.251363  111.434780 

63.275944 
66.174223 
69.159449 
72.234233 
75.401260 

70.007603 
73.457869 
77.028895 
80.724906 
84.550278 

88.509537 
92.607371 
96.848629 
101.238331 
105.781673 

77.598314 
81.702246 
85.970336 
90.409150 
95.025510 

86.163966 
91.041344 
96.138205 
101.464424 
107.030323 

95.836323 
101.62813! 
107.709546 
114.095023 

120.799774 

106.765188 
113.637274 
120.887324 
128.536127 
136.605614 

119.120867 
127.268119 
135.904206 
145.058458 
154.761966 

i  41 
42 
43 
44 
45 

78.663296 
82.023196 
85.483892 
89.048409 
92.719861 

99.826536 
104.819598 
110.012382 
115.412877 
121.029392 

112.846688]  127.839763 
118.924789]  135.231751 
125.2764041  142.993339 
131.9138421  151.14300* 
138.849969  159.700156 

145.118923  165.047684) 
154.100464  175.950545! 
103.575989  187.507577, 
173.572609  199.758032 
184.119165  212.743514! 

46 
47 
48 
49 
SO 

96.50145- 
100.39650 
104.408391 
108.54064£ 
112.79686- 

110.484031  126.870568 
115.350973  132.945390 
120.388257  139.2632W 
125.601846  145.833734 
130.997910;  152.66708- 

146.098214  168.685164  1P5.245720  226.508125 
153.672633!  178.  11942-2  20tt.!)fi4234  241.0986121 
101.5879()2!  L-r.O-jr>:!::3  21i).3(5Ktt>7  256.564529 
169.859357  l'.:f».42»iC(>3  23-2.433627,  272.958401' 
178.50302d,  209.3479'.G,  246.217477  :2',0.335'J05j 

§92.]  ANNUITIES.  295 

TABLE  III. 

THE  PRESENT  WORTH  OF  AN  ANNUITY  OF  $1.00,  FROM  1  YEAR  TO  50. 


Year. 

3  p.  cent. 

3|  p.  cent. 

4  p.  cent. 

4£  p.  cent, 

5  p.  cent. 

5^  p.  cent. 

6  p.  cent. 

Year. 

1 
2 
3 
4 
5 

0.97087 
1.91347 
2.82861 
3.71710 
4.57971 

0.961)18 
1.89969 
2.80164 
3.67308 
4.51505 

0.96154 
1.88609 
2.77509 
3.62990 
4.45182 

0.95694 
1.87267 
2.74896 
3.58753 
4.38998 

0.95238 
1.85941 
2.72325 
3.54595 

4.32948 

0.94786 
1.84631 
2.69793 
3.50.il4 
4.27028 

094339 
1.83339 
2.67301 
3.46511 
4.21236 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

5.41719 
6.23028 
7.01969 
7.78611 
8.53020 

5.32855 
6.11454 
6.87396 
7.60769 
8.31661 

5.24214 
6.00205 
6.73274 
7.43533 
8.1HKJO 

5.15787 

5.89270 
6.59589 
7.26879 
7.91272 

5.07569 
5.78637 
6.46321 
7.10782 
7.72173 

4.99553 
5.68297 
6.33457 
6.95220 
7.53762 

4.91732 
5.58238 
6.20979 
6.80169 
7.36009 

7.88687 
8.38384 
8.85268 
9.29498 
9.71225 

6 

7 
8 
9 
10 

11 

12 
13 
14 
15 

9.25262 
9.95400 
10.63495 
1L.  29607 
11.93794 

9.00155 
9.66333 
10.30274 
10.92052 
11.51741 

8.76048 
9.3850? 
9.98565 
10.56312 
11.11839 

8.52892 
9.11858 
9.68285 
10.22283 
10.73955 

8.30641 
8.86325 
9.39357 

9.89864 
10.379G6 

8.09254 
8.61852 
9.11708 
9.58965 
10.03759 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

12.56110 
13.16612 
13.75351 
14.32380 
14.87747 

12.0D412 
12.65132 
13.18968 
13.701)84 
14.21240 

11.65230 
12.16567 
12.651)30 
13.13394 
13.5U033 

11.23401 
11.70719 
12.  1591)9 
12.511329 
13.00794 

10.83777 
11.27407 

11.0895!) 
12.08532 
12.46221 

10.46216 
10.86461 
11.24607 
11.00705 
11.95034 

10.10589 
10.47726 
1082760 
11.15812 
11.46992 

16 
17 

18 
19 
20 

21 
oo 

23 
24 

25 

15.41502 
15.9361)2 
16.44361 
16.93554 
17.41315 

14.69797 
15.16712 
15.62041 
16.05637 
16.48151 

14.02916 
14.45112 

14.85684 
15.24696 
15.6220s 

13.40472 
13.78442 
14.14777 

14.49548 
14.82821 

12.82115 
13.16300 
13.48857 
13.79864 
14.0931)4 

12.27524 
12.58317 
12.87504 
13.15170 
13.413i)l 

11.76408 
12.04158 
12.30338 
12.55036 
12.78336 

21 
22 
23 
24 
25 

28 

27 
23 
29 
30 

17.87684 
18.32703 
18.76411 
19.1H845 
19.60044 

16.89035 
17.2853(1 
17.66702 
18.03577 
18.39205 

15.98277 
16.321)59 
16.0630H 
10.98371 
17.29203 

15.14661 
15.45130 
15.74287 
10.0218H 

16.28889 

14.37518 
14.64303 
14.89813 
15.14107 
15.37245 

13.G6250 
13.89810 
14.12142 
14.33310 
14.53375 

13.00317 
13.21053 
13.40616 
13.51)072 
13.76483 

13.92909 
14.08404 
14.23023 
14.36814 
14.49825 

26 
27 
28 
29 
30 

31 
32 
33 
34 

35 

20.00043 
20.38877 
20.76579 
21.13184 
21.48722 

18.73623 
19.06887 
19.39021 
19.70068 
20.00066 

17.58849 
17.87355 
18.14765 
18.41120 
18.66461 

16.54439 
16.78889 
17.02286 
17.24676 
17.46101 

15.59281 
15.80268 
10.00255 
16.192HO 
16.37419 

14.72393 
14.90420 
15.07507 
15.23703 
15.39055 

31 
32 
33 
34 
35 

36 

37 
38 
39 
40 

41 
42 
43 
44 
45 

46 
47 

48 
49 
50 

21.83225 

22.16724 
22.49246 
22.80822 
23.11477 

20.29049 
20.57053 
20.8410:) 
21.10250 
21.35507 

18.90H2H 
19.14258 
19.36786 
19.58448 
19.79277 

17.66004 

17.86224 
18.041)9!) 
18.22!)65 
18.40158 

16.54685 
16.7112!) 
10.80789 
17.01704 
17.15900 

15.53607 
15.07400 
15.80474 
15.'J2c06 
16.04612 

16.15746 
16.2021IH 
16.36303 
16.45785 
16.54772 

14.02099 
14.73078 
14.84602 

14.1)41:07 
15.04030 

36 
37 
38 
39 
40 

23.41240 
23.70136 
23.98100 
24.25427 
24.51871 

21.59910 

21.83488 
22.06269 
22.28279 
22.49545 

19.99305 
20.18563 
20.37079 
20.54884 
20.72004 

18.5061] 
18.72355 
18.87421 
19.01838 
19.15635 

17.29437 
17.  4  2321 
17.545;)! 
17.06277 
17.77407 

15.13F02 
15.22454 
15.30017 
15.38318 
15.45583 

41 
42 
43 
44 

45 

24.77515   22.700r>2 
25,02471   22.89943 
25.2ti671   23.09124 
25.50166   23.27656 
25.729761  23.45562 

20.88465 
21.042!)4 
21.19513 
21.34147 
21.48218 

1D.28837 
19.41471 
19.53561 
19.65130 
19.76201 

17.88007 
17.98102 
18.07716 
18.16872 
18.25593 

16.63283 
16.71357 
19.7D011 
16.8626l;. 
16.93143 

15.52437 
15.58903 
15.65003 
15.70757 
15.^6186 

46 
47 
48 
49 
50 

296  PROGRESSION,    OR    SERIES.  [ART.  XIX. 

Nearly  all  the  questions  that  occur  in  annuities,  can  be 
easily  solved  by  the  first  method  of  Analysis,  given  in  Sec 
tion  57,  p.  190.  This  will  be  readily  seen,  by  comparing 
the  following 

RULES. 

1.  To  find  the  amount  due  on  an  annuity  that  lias  re 
mained  unpaid  for  a  given  time. 

Multiply  the  amount  of  an  annuity  of  ONE  dollar  for  the 
given  time,a  by  the  NUMBER  of  dollars  in  the  given  annuity. 

2.  To  find  the  present  wortli  of  an  annuity  certain. 

Multiply  the  present  worth  of  an  annuity  of  ONE  dollar 
for  the  given  time,b  by  the  NUMBER  of  dollars  in  the  given 
annuity. 

3.  To  find  the  present  worth  of  a  perpetual  annuity. 

Divide  the  annuity  by  the  interest  that  ONE  dollar  would 
yield,  in  the  time  that  elapses  between  the  several  payments 
of  the  annuity. 

4.  To  find  the  annuity,  when  the  present  worth,  or  the 
amount  is  given. 

If  the  present  worth  is  given,  divide  by  the  present 
worth  of  an  annuity  of  ONE  dollar  for  the  time  the  annuity 
is  to  continue.  If  the  amount  is  given,  divide  by  the  amount 
of  an  annuity  of  ONE  dollar. 

5.  To  find  the  present  worth  of  an  annuity  in  reversion. 
Multiply  the   present  worth  in  reversion,  of  an  annuity 

*  This  amount  may  be  taken  from  Table  II. ,  p.  294,  or  it  may  be 
determined  by  finding  the  sum  of  a  geometrical  progression,  (Ex.  7, 
sect.  89,)  in  which  a  =  the  annuity,  r  —  the  amount  of  $1  for  the  time 
that  should  elapse  between  two  successive  payments,  and  7i=the 
number  of  payments  due. 

b  The  present  worth  may  be  taken  from  Table  III.  p.  295,  or  it  may 
be  found  by  dividing  the  amount  of  an  annuity  of  $1  for  the  given  time, 
by  the  amount  of  $1  at  compound  interest,  for  the  same  time. 


§92.]  ANNUITIES.  297 

of  ONE  dollar/  by  the  NUMBER  of  dollars  in  the  given 
annuity. 

EXAMPLES. 

1.  An  estate  that  yields  an  annual  income  of  $2000,  is 
offered  for  sale  for  the  amount  of  10  years'  income  at  6  per 
cent,  compound  interest.     What  is  the  price  of  the  estate  ? 

Am.  $26361.59. 

2.  A  gentleman  wishes  to  present  his  estate  to  his  chil 
dren,  reserving  enough  to  yield  $700  per  annum  for  15 
years.     How  much  must  he  reserve,  allowing  5  per  cent, 
compound  interest  ?  Ans.  $7265.76. 

3.  For  how  much  should  an  estate  that  rents  for  $175 
per  year,  be  sold,  to  allow  the  purchaser  6  per  cent,  interest 
on  his  investment  ?  Ans.  $29161 . 

4.  What  sum  of  money  must  a  man  lay  up  annually,  to 
amount  to  $10000  in  20  years,  the  investments  being  all 
made  at  6  per  cent,  compound  interest?    Ans.  $271.85. 

5.  A  father  leaves  an  annual  rent  of  $400  to  his  eldest 
child  for  5  years,  and  the  reversion  of  it  for  the  8  succeeding 
years  to  his  youngest  child.     What  is  the  present  worth 
of  each  legacy,  at  6  per  cent.  ? 

Ans.  $1684.94;  $1856.13. 

6.  If  a  person  saves  $250  per  annum,  and  invests  it  at  6 
per  cent,  compound  interest,  how  much  will  he  be  worth  at 
the  end  of  25  years?  Ans.  $13716.13. 

7.  What  sum  invested  at  6  per  cent,  compound  interest, 
will  yield  me  an  income  of  $1600  per  annum  for  25  years  ? 

Ans.  $20453.38. 


a  This  value  may  be  found  from  Table  III.,  p.  295,  by  subtracting 
the  present  worth  of  an  annuity  continuing  until  the  reversion  com 
mences,  from  the  present  worth  of  an  annuity,  continuing  until  the 
reversion  terminates.  Thus,  the  present  worth  of  an  annuity  of  $1,  to 
commence  in  3  years,  and  continue  8  years,  computing  interest  at  6 
per  cent.,  is  $7.88687  —  $2.67301  =  $5.21386. 


298  PROGRESSION,    OR    SERIES.  [ART.  XIX, 

8.  What  sum  will  build  a  wall  worth  $1000,  and  renew 
it  every  15  years,  at  5  per  cent,  compound  interest  ? 

N.  B.  At  5  per  cent,  compound  interest,  §1  in  15  years  will 
amount  to  $2.078928.  The  interest  is  therefore  $1.078028.  The 
amount  necessary  to  renew  the  wall,  is  $1000-1-1.078928,  to 
which  must  be  added  the  §1000  expended  for  building  the  wall  at 
first. 

Ans.  $1926.85. 

9.  A  builder  takes  a  lease  of  a  lot  of  ground  for  25  years, 
and  erects  buildings  on  it  which  cost  him  $20000.     Allow 
ing  money  to  be  worth  6  per  cent,  compound  interest,  what 
dcm*  annual  rent  must  he  receive  from  the  buildings  to 
reimburse  his  expenditure,  at  the  termination  of  the  lease, — 
the  rent  commencing  one  year  after  the  lease  is  given  ? 

Ans.  $1593.58. 

10.  What  sum  must  be  paid,  allowing  6  per  cent,  com 
pound  interest,  to  extend  a  lease  7  years, — the  clear  annual 
rent  being  $500,  and  the  lease  having  4  years  to  run  ? 

Ans.  $'2210.88. 

11.  What  is  the  amount  of  a  pension  of  $400  a  year, 
payable  semi-annually,  for  3  years  and  6  months,  at  7  per 
cent,  per  annum  ?  Ans.  $1555.88. 

12.  What  is  the  par  value  of  an  annual  income  of  £500 
in  the  4  per  cent,  consols  ?b  Ans.   £12500. 

13.  A  railroad  has  been  constructed  through  a  farm,  in 
consequence  of  which,  the  owner  of  the  estate  is  obliged  to 
expend  $400  in  fencing,  that  must  be  renewed  at  the  expi 
ration  of  every  12  years.     What  sum  should  he  now  receive, 
to  compensate  him  for  the  required  expenditure,  money  being 
worth  6  per  cent,  compound  interest  ?        Ans.  $795.18. 

a  The  dear  annual  rent,  is  the  amount  received  after  deducting 
ground-rent,  taxes,  and  other  expenses. 

b  CONSOLS  is  an  abbreviation  for  the  consolidated  annuities  of  the 
British  National  Debt. 


§92.]  ANNUITIES.  299 

14.  Tho  executors  of  an  estate  wish  to  dispose  of  an  un- 
expired  lease  that  has  8  years  to  run,  for  a  premium   of 
31500.     What  amount  must  be  added  to  the  annual  rent, 
for  that  purpose?  Ans.  $241.55. 

15.  What  is  the  present  worth  of  a  reversion  of  $700 
per  annum,  to  commence  in  20  years,  and  continue  30  years 
thereafter,  allowing  6  per  cent,  compound  interest  ? 

Ans.  $3004.36. 

16.  The  British  National  Debt  is  about  £800000000.    If 
£8000000  were  applied  annually  to  the  reduction  of  this 
debt,*  in  what  time  would  it  be  paid  off,  calculating  com 
pound  interest  at  the  rate  of  5  per  cent.  ? 

Ans.   36yr.  lOmo.  lody. 

17.  There  are  two  adjoining  farms,  each  renting  for  $400 
per  annum,  but  the  rent  of  one  is  payable   semi-annujilly, 
the  rent  of  the  other  quarterly.     What  will  be  the  differ 
ence  in  the  income  from  the  two  farms,  at  the  expiration 
of  20  years,  provided  all  the  rent  is  invested  as  fast  as  it 
becomes  due,  at  6  per  cent,  compound  interest  ?b 

Ans.  $190.67. 

18.  An  estate  is  sold  for  $50000,  of  which  $5000  is  to  be 
paid  in  cash,  and  the   rest  in   semi-annual  instalments  of 
$2250.     But  the  purchaser  proposing  to  discharge  the  whole 
debt  at  once,  he  wishes  to  know  what  sum  of  money  will  be 
required,  allowing  discount  at  the  rate  of  7  per  cent,  com 
pound  interest.  Ans.  $36977.90. 

19.  A  gentleman  takes  a  lease  for  ten  years,  at  $450  per 
annum.     At  the  expiration  of  two  years,  he  wishes  to  give 
up  the  lease,  but  the  landlord  will  not  consent,  unless  the 
tenant  will  either  pay  down  a  year's  rent  in  advance,  or  $60 

a  The  interest  is  supposed  to  be  regularly  paid,  in  addition  to  the 
sinking  fund  for  the  reduction  of  the  debt. 

b  The  amount  of  80  quarterly  payments  of  $1  each,  would  be 
$152.7092. 


300  POSITION.  [ART.  xx. 

per  annum,  during  the  whole  term  of  the  contract.     Which 
proposal  is  the  more  favorable,  and  how  much  will  he  save 
by  accepting  it,  money  being  worth  6  per  cent,  per  annum  ? 
2d  Ans.  He  will  save  $77.41. 

20.  What  is  the  difference  in  value,  between  the  present 
worth  of  a  lease,  for  100  years,  of  an  estate  that  rents  for 
$1500  per  annum,  and  the  perpetuity  of  the  same  estate, 
computing  interest  at  7  per  cent.  ? 

Ans.  $24.69,  or  less  than  one  week's  rent. 

N.  B.  The  present  worth  of  an  annuity  of  §1,  to  continue  100 
years,  at  7  per  cent.,  is  $14.269251. 


XX.    POSITION. 

THE  answers  to  many  difficult  analytical  questions,  can 
be  obtained  by  assuming  one  or  more  numbers,  and  working 
with  them  as  if  they  were  the  true  numbers  sought.  The 
method  of  obtaining  the  correct  result  in  such  cases,  is  called 
POSITION. 

SINGLE  POSITION  requires  only  one  assumed  number.  It 
is  used  in  solving  questions  in  which  the  required  number 
is  increased  or  diminished  by  any  of  its  parts  or  multiples, 
either  by  addition,  subtraction,  multiplication,  or  division. 

DOUBLE  POSITION  requires  two  assumed  numbers.  It  is 
applicable  to  all  questions  that  can  be  solved  by  Single  Po 
sition,  and  to  nearly  all  questions  that  can  be  solved  by 
algebraical  equations. 

93.    SINGLE  POSITION. 

Questions  in  Single  Position  may  be  solved  by  the  follow 
ing  rule : 

Assume  any  convenient  number,  and   proceed  with  it 


§94,]  DOUBLE   POSITION.  301 

according  to  the  conditions  of  the  question.  Multiply  the 
assumed  number  by  the  number  given  in  the  question,  and 
divide  the  product  by  the  result  obtained  with  the  assumed 
number. 

EXAMPLE  FOR  ILLUSTRATION. 

Divide  $1584  among  three  persons,  in  such  manner  that  J  of 
the  first  share,  J  of  the  second  share,  and  £  of  the  third  share, 
shall  all  be  equal. 

Suppose  J  the  first  share  250. 

Then  the  first  share  will  be    500  250  X  1584-:-  2250  =  176, 

second  share  750  is  %  the  first  share. 

third  share  1000  Ans.  1st  share  $352 

2d      "     $528 
Result        2250  3d      « 


Solve  by  the  above  rule,  Examples  8,  15,  22,  23,  25,  28, 
29,  30,  32,  33,  42,  46,  47,  58,  59,  64,  71,  73,  74,  79,  81, 
Section  60. 

O4.    DOUBLE  POSITION. 

RULE  I. 

Assume  two  convenient  numbers,  proceed  with  them  sepa 
rately,  according  to  the  conditions  of  the  question,  and  note 
the  result  obtained  from  each  operation. 

Multiply  the  error  of  either  result  by  the  difference  of 
the  assumed  numbers,  and  divide  the  product  by  the  differ 
ence  of  the  results.  The  quotient  will  be  a  correction  to 
be  added  to  the  assumed  number,  11  :t,  gives  a  result  too 
small,  or  to  be  subtracted  from  it,  if  the  result  is  too  large. 

In  many  cases  the  correct  result  is  obtained  at  the  first  trial, 
but  if  greater  accuracy  is  required,  take  the  number  obtained  by 
the  first  trial,  and  the  nearer  of  the  two  numbers  first  assumed, 
or  any  other  that  appears  more  nearly  correct,  as  new  assumed 
numbers.  Repeat  the  operation  as  before,  and  you  will  obtain  a 
new  answer,  more  accurate  than  the  former.  This  process  may 
be  repeated  until  you  obtain  the  true  answer,  or  a  number  suf 
ficiently  correct  for  your  purpose. 

This  Rule  fails  in  those  questions  in  which  the  result  of  the 
operations  to  be  performed  is  not  a  known  number,  but  the  re- 


302  POSITION.  [ART.  XX. 

quired  number,  or  one  depending  upon  it,  such  as  some  multiple, 
or  some  part  of  it.  Such  cases  may  usually  be  solved  by  the  fol 
io  wing  rule. 

RULE  II. 

Assume  two  convenient  numbers,  and  perform  on  each 
of  them  the  operations  indicated  by  the  question.  Note 
the  errors  of  the  results,  and  mark  each  of  them  with  the 
sign  +  or  —  ,  according  as  it  is  in  excess  or  defect. 

Then  form  the  following  proportion  :  — 

The  difference  of  the  errors,  when  they  are  alike*  or  their 
sum,  when  they  are  unlike  :  the  difference  of  the  assumed 
numbers  :  :  either  error  :  the  correction  of  the  assumed  num 
ber  which  produced  that  error. 

EXAMPLES  FOR  ILLUSTRATION. 

1.  "A  hunter  wishes  to  measure  the  width  of  a  ravine,  and 
having  no  other  means  of  doing  it,  he  fires  a  rifle  ball  at  a  small 
knot  on  a  tree,  which  stands  on  the  opposite  bank.  On  going 
over,  he  finds  that  the  ball  struck  6  inches  below  the  knot.  By 
previous  experiments,  he  knows  that  the  ball  drops  2  inches  in 
going  50  rods,  and  its  deflection  is  proportional  to  the  square  of 
the  distance  added  to  £  of  its  cube.  What  is  the  width  of  the 
ravine  ?" 

Suppose  100 


(W/  +  J  of  (W  X  2  =  133  first  result. 

Subtract         6 
Error  in  excess         7£ 
Suppose  75 

(U)*+  3°f  Gfi-)3*  2  =  6f  sccond  result- 

Subtract         6 
Error  in  excess          f 

Then  by  Rule  I,  7^  X  25  -r  6^  —  27f|,  correction  for  first 
supposition.  100  —  27^|  —  72^|  rods,  approximate  width. 

Working  with  this  result,  we  obtain  for  the  deflection  6.168 
inches,  which  is  so  near  the  true  deflection  that  72  rods  may  be 
assumed  as  the  true  width. 

a  The  errors  are  said  to  be  alike,  when  both  are  too  great,  or  both 
too  small  ;  and  unlike,  when  one  is  too  great,  and  the  other  too  small. 


§94.]  DOUBLE   POSITION.  303 

2.  A  person  being  asked  the  time,  replied  :  "  The  sun  now  rises 
at  5  and  sets  at  7.  Now  if  you  add  }  of  the  hours  that  have 
passed  since  sunrise,  to  -£-f  of  those  which  must  elapse  before 
sunset,  you  will  have  the  exact  time  of  the  day." 

To  find  how  many  hours  have  elapsed  since  sunrise,  by  Rule  II. 
Suppose  6.  Suppose  3. 

$  +  Jfof  8  =  6|f  f  +  Jf  of  11  =  8^- 

The  result  should  be  11.  The  result  should  be  8. 

1st  Error  —  4^-.  2d  Error  -f  '  }. 

4^f  :  3  : :  4^r  :  2|  correction  to  be  subtracted  from  6. 
6—  2£=3£  6  +  3£  =  8£  the  true  time. 

Proof.     ^  of  3j  +  -Jfof  10i=8i, 

EXAMPLES  FOR  THE  PUPIL. 

1.  A  farmer  engaged  a  servant,  agreeing  to  pay  him  $1.50 
for  every  day  he  should  work,  and  to  charge  him  $.50  per 
day  for  his  board,  every  day  he  should  be  idle.     At  the  end 
of  13  weeks,  the  man  received  $86.50.     How  many  days 
did  he  work  ?  Ans.  66  days. 

2.  Two  men  have  the  same  income.     A.  saves  J  of  his, 
but  B.  spends  $325  per  year  more  than  A.,  and  at  the  end 
of  5  years  finds  himself  $625  in  debt.     What  is  the  annual 
income  of  each  ?  Ans.  $1000. 

3.  A  person  distributed  in  charity  2d.  apiece  among  several 
poor  children,  and  had  4d.  left.     He  would  have  given  them 
3d.  apiece,  but  had  not  enough  money  by  lOd.     What  was 
the  number  of  children  ?  Ans.  14. 

4.  A  man  has  a  chaise  worth  $130,  and  two  horses.     If 
the  first  horse  be  harnessed  to  the  chaise,  their  joint  value 
will  be  3  times  that  of  the  second  horse ;  but  if  the  second 
horse  be  harnessed  to  the  chaise,  their  joint  value  will  be 
twice  that  of  the  first  horse.     Required  the  value  of  each 
horse.  Ans.  First  horse  $104;  second  $78. 


304  POSITION.  [ART.  XX. 

5.  Find  two  numbers,  whose  difference  is  29,  and  their 
product  546.  Ans.  13  and  42. 

6.  A  man  sold  a  horse  for  $144,  and  thereby  gained  as 
much  per  cent,  as  was  equivalent  to  the  number  of  dollars 
that  the  horse  cost  him.     How  much  did  he  give  for  the 
horse?  Ans.  $80. 

7.  The  fourth  power  of  a  certain  number,  diminished  by 
7  times  the  number,  and  increased  by  3  times  its  cube, 
equals  3381.     What  is  the  number  ?  Ans.  7. 

8.  Find  a  number  to  which  if  7  times  its  square  be  added, 
the  sum  will  be  500.  Ans.  8.3804 +  . 

9.  John  is  now  three  times  as  old  as  Charles,  but  five 
years  ago  he  was  four  times  as  old.     Required  the  age  of 
each.  Ans.  John  45;  Charles  15. 

10.  Five  times  a  certain  number,  increased  by  12,  is 
equivalent  to  7  times  the  number,  diminished  by  20.    What 
is  the  number  ?  Ans.  16. 

11.  What  number  is  that  whose  half  is  as  much  less 
than  75,  as  its  double  is  greater  than  94  ?         Ans.  67|. 

12.  A  farmer  purchased  a  number  of  geese  for  £6  5s. 
He  retained  5,  and  sold  the  remainder  for  Is.  3d.  apiece 
more  than  he  paid,  thus  receiving  what  he  paid  for  the 
whole.     How  many  did  he  buy  ?  Ans.  25. 

13.  The  area  of  a  certain  field  is  187  square  rods,  and 
the  length  exceeds  the  breadth  by  6  rods.     What  are  the 
dimensions  ? 

14.  There  is  a  fish,  whose  head  weighs  9  pounds ;  his 
tail  weighs  as  much  as  his  head  and  half  his  body ;  and 
his  body  weighs  as  much  as  his  head  and  tail  both.     What 
is  the  weight  of  the  fish  ?  Ans.  72  Ib. 

15.  What  would  have  been  the  width  of  the  ravine,  in 
the  example  given  on  p.  302,  if  the  ball  had  struck  8  inches 
below  the  knot  ?  Ans.  801  rods,  nearly. 


§95]  MULTIPLICATION.  305 

XXI.    APPBOXIMATIONS. 
95.    MULTIPLICATION. 

In  MULTIPLICATION,  if  only  a  certain  degree  of  accuracy  is 
deaired,  the  product  may  be  obtained  by  writing  the  units'  figure 
of  the  multiplier  under  that  figure  of  the  multiplicand  whose 
place  we  would  reserve  in  the  product,  and  inverting  the  order 
of  the  remaining  figures.  In  multiplying,  we  commence,  for  each 
partial  product,  with  the  figure  of  the  multiplicand  immediately 
above  the  multiplying  figure,  carrying  the  tens  which  would  arise 
from  the  multiplication  of  the  two  rejected  figures  at  the  right. 

EXAMPLE. 

Required  the  product  of  287.613952  by  15.98421,  correct  to 
the  fourth  decimal  place. 

287.613952  287.613952 

12489.51  15.98421 


2876.1395  28 

1438.0698  575 

258.8525  11504 

23.0091  230091 

1.1505  2588525 

575  14380697 

29  28761395 


7613952 

227904 

55808 

1616 

568 

60 


4597.2818  4597.28180769792 

The  units'  figure  of  the  multiplier  being  placed  under  the  4th 
decimal  of  the  multiplicand,  and  the  whole  multiplier  reversed, 
the  product  of  each  figure  by  the  one  above  it  will  be  ten-thou 
sandths.  Therefore,  the  right-hand  figure  of  each  partial  product 
will  fall  in  the  column  of  ten-thousandths.  In  the  second  product, 
multiplying  52  by  5,  we  obtain  260,  which,  being  nearer  300  thau 
200,  we  carry  3  to  the  product  of  9  by  5. 

The  multiplication  has  also  been  performed  in  the  usual  way, 
the  vertical  line  showing  the  figures  that  are  rejected. 

If  the  multiplicand  does  not  contain  enough  decimal  figures  to 
correspond  with  the  inverted  multiplier,  the  deficiency  should  be 
supplied  by  annexing  zeros.  The  same  contraction  may  be 
applied  to  integers,  if  we  wish  only  to  obtain  the  thousands, 
millions,  &c.,  of  the  product. 

20 


306  APPROXIMATIONS.  [ART.  XXI; 


9G«    DIVISION. 

In  DIVISION,  a  similar  contraction  may  be  made  when  the 
divisor  is  large,  a  contraction  which  is  also  applicable  in  the 
extraction  of  roots. 

The  first  quotient  figure  is  of  the  same  numerical  value  as  the 
figure  of  the  dividend  which  stands  immediately  over  the  units  of 
the  divisor,  at  the  first  step  of  the  division. 

After  the  first  remainder  has  been  obtained,  instead  of  bringing 
down  the  remaining  figures  of  the  dividend,  we  may  cut  off  the 
right-hand  figure  of  the  divisor  at  each  step,  as  in  the  following 
example. 

342.15 )  28417.95255  ( 83.057  342.15 )  28417.9 1 5255  (  83.057 
27372  0  27372  0| 

10459  1045915 

10264  10264  5 


195  1951025 

171  1711075 


24  23 

24  23 


9505 
9505 


In  the  complete  division,  the  contraction  is  indicated  by  the 
vertical  line.  In  each  multiplication,  the  tens  arising  from  the 
product  of  the  quotient  figure  by  the  suppressed  figure  of  the 
divisor,  must  always  be  carried  as  in  contracted  multiplication. 

The  right-hand  figure  of  the  quotient  thus  obtained,  cannot 
always  be  relied  upon.  If  greater  accuracy  is  desired,  the  divi 
sion  may  be  extended  further  before  commencing  the  contraction. 

IN  DIVISION  OF  CIRCULATING  DECIMALS,  we  may  adopt  the  fol 
lowing  rule. 

Make  the  repetends  of  the  divisor  and  dividend  similar  and 
conterminous,  and  from  the  result,  considered  as  whole  numbers, 
subtract  the  finite  part  of  each.  Perform  the  division  with  the 
remainders  as  with  whole  numbers,  and  the  true  quotients  will 
be  obtained. 

EXAMPLE. 

Divide  36.91  by  5.273. 


§97.]  CONTINUED   FRACTIONS.  307 

The  example  is  here   solved  5.273273)36.919191 
by  contracted  decimal  division.  5  86 

The  exact  fractional  quotient  is  5273268)  3691915577.001191 

7««rWo9*«  or  7T-o3Q3Tzr.     The  ef-        36912876 


feet    of    subtracting    the   finite  6279 

parts  of  the  divisor  and  dividend  5273 

is  the  same  as  reducing  the  two  1006~ 

numbers  to  improper  fractions,  527 

and  dividing  the  numerators. 

47  «7 
474 

~T 
5 

97.    CONTINUED  FRACTIONS. 

CONTINUED  FRACTIONS  arise  from  the  approximate  valuation 
of  fractions  whose  terms  are  large,  and  prime  to  each  other.  If, 
for  example,  we  desire  approximate  values  for  the  fraction  48g9^j 
we  may  commence  by  dividing  both  terms  of  the  fraction  by  the 

numerator,  which  gives  us  5^2  •    Disregarding  the  ||,  we  have  -J 

53 

for  a  first  approximate  value,  which  is  greater  than  the  true  value, 
because  the  approximate  denominator  is  less  than  the  true  denomi 
nator.  But  as  the  denominator  is  between  5  and  6,  the  fraction 
is  between  -|-  and  |- 

If  we  desire  greater  accuracy,  we  may  divide  f  f  in  the  same 
manner  as  the  first  fraction,  which  gives  us  ___  for  the  value  of 
4|,  or 


for  a  second  value  of  the  original  fraction.     Disregarding  the 
452,  the  continued  fraction  becomes, 


or  YJ,  which  is  less  than  the  true  value,  because  the  supposed 
denominator  is  greater  than  the  true  denominator.  We  therefore 
know  that  the  fraction  is  between  5  and  -jfy. 


308  APPROXIMATIONS.  [ART.  XXI. 

Still  greater  accuracy  may  be  obtained  by  reducing  4%  which 
gives  us 


Rejecting  the  f ,  we  have, 
1 

J_,      or  E, 
6A  93 

for  a  third  approximate  value,  greater  than  the  true  value.     The 
fraction  is,  therefore,  between  -g-|  and  ^j. 

After  one  farther  approximation,  we  should  obtain  the  original 
fraction.  In  fractions  whose  terms  are  very  large,  as  in  the  ratio 
of  the  diameter  to  the  circumference  of  a  circle,  these  approximate 
values  are  often  very  useful.  They,  moreover,  have  the  advantage 
of  admitting  any  required  degree  of  accuracy,  for  the  error  in 
adopting  any  approximation,  is  always  less  than  the  difference 
between  the  fraction  taken  and  the  one  following.  Thus,  in  the 
present  example,  if  we  had  adopted  5  as  the  true  value  of  ^gHy, 
the  error  would  have  been  less  than  §  —  g,  or  3*3. 

For  forming  the  successive  approximations,  we  have  the  fol 
lowing 

RULE. 

Divide  the  greater  term  by  the  less,  and  the  divisor  by  the 
remainder,  &c.,  as  in  finding  the  greatest  common  measure. 

Assume  1  for  the  numerator,  and  the  first  quotient  for  the  de 
nominator  of  the  first  approximate  value. 

Multiply  the  terms  of  this  fraction  by  the  second  quotient,  and 
add  1  to  the  product  of  the  denominator,  for  the  second  approxi 
mate  value. 

For  each  succeeding  approximation,  multiply  the  terms  of  the 
last  approximate  fraction  by  the  following  quotient,  and  add  the 
corresponding  terms  of  the  preceding  fraction. 

If  the  fraction  given  is  improper,  the  reciprocals  of  the  fractions 
thus  obtained,  will  be  the  approximations  desired. 

EXAMPLE. 

Required  less  approximate  values  for  the  ratio  of  the  circum- 


§97.]  CONTINUED  FRACTIONS.  309 

ference  to  the  diameter  of  a  circle,  one  approximate  ratio  being 

WWW- 

100000)314159(3  Equivalent  Continued  Fraction. 


14159)100000(7 


99113  7  + 


887)14159(15 
887 


15  + 


25  + 


5289  1  + 1_ 

4435  7£ 

864)887(1 

854  For  convenience,  the  frac- 

33)854(25    tion  mav  be  written, 
i    i    i i_  _i_  i_i 

'7+15+1+25+1+7+4- 


194 
165 

29)88(1 

29 

4)29(7 

28 

1)4(4 

4 
1 

1st  approximate  value. 

1X7  =7 

~o  v     7-1-     1=22         approximate  value. 

7  x  15  +    1  =  106 

22  x  15  +    3  ="333   3d  approximate  value. 
106  X     1  +    7  =  113 
"333  X     1  +  22  ="355  4th  approximate  value. 

&c.,  &c. 

The  reciprocals  of  these  values  are, 

3,  2?,   ?!!,   and  3A5. 
7      106  113 

The  second  ratio  is  the  one  given  by  Archimedes.  The  fourth 
is  that  of  Adrian  Metius,  and  is  even  more  exact  than  the  ratio 
3.14159,  from  which  we  have  derived  it. 

An  infinite  continued  fraction  may  be  made  equivalent  to  any 
given  number  by  the  following  rule  : 

Assume  any  number  you  please  for  a  denominator;  add  the  assumed 


310  APPROXIMATIONS.  [ART.  XXI. 

number  to  the  given  number,  and  multiply  the  sum  by  the  given  number, 
for  a  numerator. 

_          40     40    40   .     0  666 

EXAMPLES.      5  =  r+  ^  3-+  ,  2  =  -  -  -. 

The  value  of  any  infinite  continued  fraction,  with  but  one 
numerator  and  denominator,  may  be  found  by  the  following 
rule: 

To  4  times  the  numerator  add  the  square  of  the  denominator.  From 
the  square  root  of  the  sum,  subtract  the  denominator,  and  divide  the 
remainder  by  2.  The  quotient  is  the  value  sought. 


EXAMPLES.    |£  |J  =  (v/160"+^—  3)^2  =5; 

££=  (Vl2+16-4)-,-2  =  .6457+ 
The  SQUARE  ROOT  of  any  number  may  be  expressed  in  the  form 
of  a  continued  fraction,  after  part  of  the  root  is  found,  —  by  malting 
each  numerator  equal  to  the  remainder,  and  each  denominator  equal  to 
twice  the  root  found.  Thus,  in  extracting  the  square  root  of  17,  the 
first  root  figure  is  4,  and  the  remainder  1.  Then  the  true  root  is 
4  -\-  the  continued  fraction 

11111     p 

8+  8+  8~+  8+  8+?   &C' 

In  like  manner,  the  square  root  of  14,  is 

5      5      5      5        n 
3  +  6+6+6+  6+,   &C. 

Reducing  the  fraction,  we  have,  first, 


nearly,  giving  the  first  approximate  root  3|.     Second, 

—   205   ^   17 
630   —   275  °r  237 

nearly,  giving  a  second  approximate  root  3j|.     Third, 


nearly,  giving  a  third  approximate  root  3f  f.  This  approximation 
is  of  use  in  affording  convenient  fractional  expressions  for  those 
roots  which  are  of  most  frequent  occurrence.  Thus,  the  diagonal 
of  a  square  is  to  its  side  as  \/2~is  to  1.  By  the  rule  just  given, 
we  obtain  successively  for  approximate  values  of  \/2, 

11»  1I.  JA.  41.  J§  *• 

Ihe  last  of  these  values,  if  §  or  f  j,  is  a  very  convenient  one. 


§  98.]  EVOLUTION.  311 


98.    EVOLUTION. 

In  the  Extraction  of  Roots,  we  may  commence  with  any  com 
plete  divisor,  cutting  off  the  right  hand  figure  at  each  step,  as  in 
contracted  division.  At  whatever  place  this  contraction  is  com 
menced,  as  many  additional  root  figures  will  be  obtained  as  are 
equal  to  the  number  of  figures  in  the  divisor  less  1,  but  the  last 
figure  so  obtained  cannot  be  relied  upon.  To  illustrate  this  mode 
of  contraction,  we  will  extract  the  5th  root  of  69. 


0 

2 

0 
4 

0 

8 

0 
16 

69.(2.3323285 
32 

2 

2 

4 

8 

8 
24 

16 
64 

37 

32.36343 

4 

2 

12 
12 

32 
48 

80 
12.927 

80 

27.8781 

4.63657 
4.29498 

6 
2 

24 
16 

107.8781 
32.0424 

34159 
29334 

8 
2 

40 

3.09 

92.927 
13.881 

139.9205 
3.246- 

4825 
4407 

418 

294 

117 

10.3 
3 

43.09 
3.18 

106.808 
1.4-- 

143.166 

3.288 

10.6 

46.27 

108.2 
1.4 

146.454 
22- 

109.6 

146.67 
22 

7 
7 

146.89 

After  obtaining  the  third  trial  divisor,  we  commence  rejecting 
one  figure  from  the  trial  divisor,  two  from  the  number  at  the  foot 
of  the  preceding  column,  three  from  the  third  column,  &c.,  and 
proceed  in  a  similar  way  with  each  subsequent  trial  divisor,  until 
the  figures  from  the  preceding  columns  are  entirely  cancelled. 
But  in  every  instance,  allowance  must  be  made  for  the  product 
of  the  figures  rejected,  as  in  simple  contracted  division. 

The  following  is  a  general  rule  for  the  approximation  of  ANY 
ROOT  desired. 

RULE. 
Oall  the  first  two  figures  of  the  root  found  in  the  usual  way,  the 

ASCERTAINED  ROOT. 


312  APPROXIMATIONS.  [ART.  XXI. 

Involve  the  ascertained  root  to  the  given  power,  and  multiply 
by  the  index  of  the  root  for  a  dividend. 

Subtract  the  power  of  the  ascertained  root  from  the  correspond 
ing  periods  of  the  given  number,  for  a  divisor.  Divide,  and  reserve 
the  quotient. 

To  6  times  the  reserved  quotient,  add  the  index  of  the  root, 
plus  1,  for  a  second  dividend. 

To  6  times  the  reserved  quotient,  add  4  times  the  index  of  the 
root,  subtract  2  from  the  sum,  and  multiply  by  the  reserved  quo 
tient  for  a  second  divisor.  Divide,  add  1  to  the  quotient,  and 
multiply  by  the  ascertained  root  for  the  true  root  nearly.  If 
greater  accuracy  is  desired,  repeat  the  process  with  the  root  thus 
found. 

By  this  rule,  the  number  of  figures  in  surd  roots,  may  generally 
be  tripled  at  each  operation. 


The  following  is  the  application  of  the  rule,  in  extracting  the 
5th  root  of  659901. 

Ascertained  root  14. 

145=  537824  given  no.  659901 

index  5  145  =  537824 


dividend  2689120  1st  divisor  122077 

2689120  -:-  122077  =  22.02806,  reserved  quotient, 
reserved  quotient  22.02806 


132.16836 
index  +1       6. 


second  dividend  138.16836 
6  X  reserved  quotient  =  132.16836 
4  x  5  — 2  =  18. 


150.16836 
Multiply  by     22.02806 


2d  divisor  3307.91764 
138.16836  -f-  3307.91764  =  .041768 

1.041768  x  14  =  14.584752,  approximate  root, 
correct  to  the  fourth  decimal  place. 

This  contraction  is  of  use  in  extracting  the  higher  roots.  Any 
root  below  the  10th  may  be  obtained  in  the  usual  way,  nearly  as 
readily,  and  with  much  greater  accuracy. 


§100.]        PROPERTIES    OF   SQUARES   AND   CUBES.  313 


99.    EXAMPLES  IN  APPROXIMATION. 

1.  Multiply  11817.93642  by  2581.36,  and  reserve  two  decimal 
places. 

2.  Divide  2704.1583  by  361.8901. 

3.  Divide  4.3097  by  18.615843. 

4.  What  are  the  approximate  values  of  .785398,  •which  is  nearly 
the  ratio  of  the  area  of  a  circle,  to  that  of  its  circumscribing  square  ? 

A™-  i;  I;  f;  3,  ii;  Mi;  ii!>&°- 

5.  Form  infinite  continued  fractions,  equivalent  to  15  ;  to  7  ;  to 
20;  to  5£. 

6.  Determine  the  value  of  the  infinite  fraction  3^  3^  ^?. 

Ans.  $. 

7.  Express  in  the  form  of  a  continued  fraction  N/19;  (18)^; 
</3T. 

8.  Find  the  5th  root  of  729.  Ans.  3.73719. 

9.  Extract  the  17th  root  of  1.004.  Ans.  1.00023. 


XXII.    PROPERTIES  OF  NUMBERS/ 
1OO.    PROPERTIES  or  SQUARES  AND  CUBES. 

1.  EVERY  square  number  terminates  in  1,  4,  5,  6,  or  9,  or  in 
an  even  number  of  ciphers  preceded  by  one  of  these  figures.  If 
a  square  number  ends  in  1,  4,  5,  or  9,  the  last  figure  but  one  will 
be  even,  but  if  it  ends  in  6,  the  preceding  figure  will  be  odd.  If 
a  square  ends  in  5,  it  will  end  in  25,  and  the  figure  preceding 
25  must  be  even. 

No  square  number  can  end  in  two  even  digits,  except  two 
ciphers,  or  two  fours.  No  square  number  can  end  in  three  equal 
digits,  except  three  fours ;  nor  in  more  than  three  equal  digits, 
unless  they  are  ciphers. 

a  Hutton,  Barlow,  and  private  sources. 


314  PROPERTIES    OF    NUMBERS.  [ART.  XXII. 

2.  Every  square  number  is  divisible  by  3,  or  becomes  so  when 
diminished  by  1.     The  same  remark  may  be  made  of  4.     If  1  be 
deducted  from  any  odd  square  number,   the  remainder  will  be 
divisible  by  8.     If  a  square  be  either  multiplied  or  divided  by  a 
square,  the  product  or  the  quotient  will  be  a  square. 

3.  Every  number  is  either  a  square,  or  is  divisible  into  two, 
three,  or  four  squares. 

4.  Every  power  of  5,  or  of  any  number  terminating  in  5,  neces 
sarily  ends  in  5.     A  similar  remark  may  be  made  of  1,  and  6. 

5.  Assume  any  two  numbers  whatever ;  then  one  of  them,  or 
their  sum,  or  their  difference,  must  be  divisible  by  3. 

6.  If  the  sum  of  two  squares  forms  a  square,  the  product  of 
their  square  roots  will  be  divisible  by  6. 

7.  If  1  be  added  to  the  product  of  two  numbers  whose  differ 
ence  is  2,  the  sum  will  be  the  square  of  the  intermediate  number. 

8.  If  a  cube  be  divisible  by  6,  its  root  will  also  be  divisible  by 
6.     And  if  a  cube,  when  divided  by  6,  has  any  remainder,  its  root 
divided  by  6  will  have  the  same  remainder. 

9.  All  exact  cubes  are  divisible  by  4,  or  can  be  made  so  by  add 
ing  or  subtracting  1.     The  same  remark  may  be  made  of  7,  and  9. 

10.  The  cube  root  of  any  exact  cube,  consisting  of  not  more 
than  six  figures,  may  be  determined  by  inspection.     Divide  the 
number  into  periods,  (as  in  the  usual  mode  of  extracting  the  cube 
root,)  and  to  the  root  of  the  greatest  cube  contained  in  the  first 
period,  affix  the  root  of  the  cube  that  terminates  in  the  right  hand 
figure  of  the  second  period. 

11.  If  a  cube  terminates  in  ciphers,  the  number  of  ciphers  must 
be  divisible  by  3. 

12.  Any  square  may  be  divided  into  two  other  squares,  in  the 
following  manner : — 

Assume  any  two  numbers  at  pleasure,  and  by  their  product 
multiply  double  the  root  of  the  given  square,  for  a  numerator. 
Take  the  sum  of  the  squares  of  the  assumed  numbers  for  a  de 
nominator.  The  resulting  fraction  will  be  the  root  of  one  of  the 
squares  sought.  Subtract  the  square  of  this  root  from  the 
given  square,  and  the  remainder  will  be  the  other  square  re 
quired. 


§100.]        PROPERTIES   OF   SQUARES   AND    CUBES.  315 

13.  If  we  add  the  cubes  of  the  series,  1,  2,  3,  4,  commencing 
at  the  beginning,  and  taking  any  number  of  terms  whatever,  the 
sum  will  always  be  a  square.     Thus,  l-{-8  =  9;  1  -J-  8  -f-  27  = 
36;  1  +  8  +  27  +  64=100. 

14.  If  we  write  down  the  series  of  squares  of  the  natural  num 
bers,  and  take  the  diiference  between  the  successive  terms,  and 
the   difference   of  these  differences,   the  second  differences  will 
always  be  2,  as  may  be  seen  below : 

Squares.  1  4  9  16  25  36 

IstDiff.  357  9  11 

2dDiff.  2222 

15.  If  the  successive  differences  of  the  series  of  cubes  be  taken, 
the  third  differences  are  always  6,  =  1  x  2  X  3,  as  may  be  seen 
below : 

Cubes.  1  8  27  64  125  216 

IstDiff.  7  19  37  61  91 

2d  Diff.  12  18  24  30 

3d  Diff.  666 

16.  The  fourth  differences  of  the  series  of  fourth  powers  are 
always  equal  to  1  X  2  X  3  X  4  =  24 ;  the  fifth  differences  of  the 
series  of  fifth  powers  are  always  equal  tol  X  2  X  3x4x5  = 
120;  and  so  on. 

EXAMPLES. 

1.  By  which  of  the  foregoing  rules  do  you  know  that  neither 
of  the  following  numbers  can  be  a  square?     952;  827;  1814; 
2795;   3725;    308;    711;    866;    299;    25000;    334;    779;    426; 
47800. 

2.  Divide  24  into  three  squares.  Am.  16-J-4-J-4. 

3.  Find  four  square  numbers,  whose  sum  will  make  30. 

4.  If  you  divide  the  cube  of  8709512863  by  6,  what  will  be  the 
remainder  ? 

5.  By  what  rules  do  you  know  that  neither  of  the  following 
numbers  is  an  exact  cube  ?     87042  ;  14284  ;   730176  ;  4080000  ; 
51858. 

6.  Determine  by  inspection,  the  cube  root  of  each  of  the  fol 
lowing   exact  cubes.     12167;    21952;    103823;    39304;    42875; 
97336;  79507;  132651;  24389.  Ana.  23;  28;  47,  &c. 


816  PROPERTIES   OF   NUMBERS.  [ART.  XXII, 

7.  Divide  49  into  two  other  squares. 

^.Assuming  2  and  3;  f  *2!ii)' =  !55?  ,  49-7056=l^, 
V4+9-  169  169          169  > 

Assuming  1  and  8; 

&c.  &c.  &c. 

1O1.     PRIME  AND  COMPOSITE  NUMBERS. 

Every  number  that  cannot  be  divided  by  any  other  number, 
(except  1,)  without  a  remainder,  is  called  a  PRIME  NUMBER. 

Two  or  more  numbers  that  have  no  common  divisor,  are  said  to 
be  prime  to  each  other.  Every  prime  number  is  prime  to  all  other 
numbers  except  its  own  multiples. 

There  are  no  known  means  of  determining  at  once  whether  a 
proposed  number  is  a  prime ;  but  the  following  properties  and 
rules  will  enable  us  to  determine  all  the  divisors  of  any  number. 

1.  2  is  a  factor  of  all  numbers  terminated  by  0,  2,  4,  6,  or  8. 
For,  as  2  will  divide  10,  it  will  also  divide  any  number  of  tens,  or 
any  number  of  tens  plus  2,  4,  6,  or  8.     Numbers  divisible  by  2 
are  called  EVEN  ;  all  others,  ODD  numbers. 

2.  5  is  a  factor  of  all  numbers  terminated  by  0  or  5.     For,  as 
5  will  divide  10,  it  will  also  divide  any  number  of  tens,  or  any 
number  of  tens  plus  5. 

3.  3,  or  9,  is  a  factor  of  all  numbers  in  which  the  sum  of  the 
figures  is  exactly  divisible  by  3,  or  9.     For,  if  from  any  power  of 
10,  as  10,  100,  1000,  &c.,  we  subtract  1,  the  remainder  consists 
entirely  of  9's,   and   is,    therefore,   divisible  by  both  3   and  9. 
Hence,  any  power  of  10  is  divisible  by  3  and  9  with  1  remainder; 
therefore,  any  number  of  tens,  hundreds,  thousands,  &c.,  dimi 
nished  by  as  many  units,  will  be  divisible  by  3  and  by  9.    Let  us, 
then,  examine  the  number  34794.  3  ten  thousands  —  3  ;  4  thou 
sands —  4;   7  hundreds  —  7;  9  tens  —  9;   and  4  units  —  4;  each 
divided  by  3  or  9,  give  no  remaina<;r.     Therefore,  34794  —  3  —  4 
—  7  —  9  —  4,  is  divisible  by  3  and  by  9,  and  if  the  sum  of  the 
numbers  subtracted,  or  in  other  words,  the  sum  of  the  digits,  is 
similarly  divisible,  the  number  itself  will  be  so. 

4.  11  is  a  factor  of  all  numbers  in  which  the  sum  of  the  odd 
digits,  (the  1st,  3d,  5th,  &c.,)  and  the  sum  of  the  even  digits,  (the 
2d,  4th,  6th,  &c.,)  are  equal,  or  their  difference  is  some  multiple 
of  11.     For  any  number  of  tens,  thousands,  hundred  thousands, 


§101.]  PRIME   AND    COMPOSITE    NUMBERS.  317 

&c.,  (-which  represent  the  even  digits,)  increased  by  as  many 
units,  will  be  divisible  by  11.  Any  number  of  hundreds,  ten 
thousands,  millions,  &c.,  (which  represent  the  odd  digits,)  dimi 
nished  by  as  many  units,  will  also  be  divisible  by  11.  Take,  then, 
the  number  635173.  6  hundred  thousands  +  6 ;  3  ten  thousands — 
3 ;  5  thousands  +  5  ;  1  hundred  —  1 ;  70  +  7 ;  and  3  —  3  ;  each 
divided  by  11  give  no  remainder.  Therefore,  635173  — 18+  7  or 
635173  —  11,  is  divisible  by  11,  and  635173  itself  must  be  so. 

5.  4  is  a  factor  of  all  numbers,  in  which  the  two  terminating 
figures  are  divisible  by  4.     For,  as  4  will  divide  100,  it  will  also 
divide   any  number   of  hundreds,  or  any  number  of  hundreds 
plus  any  number  of  units  divisible  by  4. 

6.  25  is  a  factor  of  all  numbers  terminating  in  25,  50,  75,  or 
two  zeros.     For,  as  25  will  divide  100,  it  will  also  divide  any 
number  of  hundreds,   or  any  number  of  hundreds  plus  25,  50, 
or  75. 

7.  Every  number  that  is  divisible  by  two  or  more  numbers 
prime   to  each  other,  is   divisible  by  their  product.     Take,  for 
example,  105,  which  is  divisible  by  both  3  and  5.     This  number 
may  be  resolved  into  the  factors  5  X  21 ;  5  x  21,  must  therefore 
be  divisible  by  3.     But  as  3  will  not  divide  5,  it  must  divide  the 
other  factor  21,  and  the  number  may  be  resolved  into  the  factors 
5  X  3  X  7  or  15  X  7.     Hence  we  deduce  the  following  additional 
properties. 

8.  Every  even  number  that  is  divisible  by  3  is  also  divisible  by 
6 ;  and  every  even  number  that  is  divisible  by  9  is  also  divisible 
by  18. 

9.  Every  number  divisible  by  3  or  9,  in  which  the  two  termi 
nating  figures  are  divisible  by  4,  is  divisible  by  12  or  36. 

10.  Every  number  divisible  by  3  or  9,  whose  terminating  digit 
is  0  or  5,  is  divisible  by  15  or  45. 

11.  Every  prime  number  greater  than  2,  is  one  greater  or  one 
less  than  some  multiple  of  4. 

12.  Every  prime  number  greater  than  3,  is  one  greater  or  one 
less  than  some  multiple  of  6. 

13.  Every  number  that  has  no  prime  factor,  equal  to  or  less 
than  its  square  root,  is  itself  a  prime  number.     For  the  product 
of  any  two  factors,  each  greater  than  the  square  root  of  a  number, 
would  evidently  be  greater  than  the  number  itself.     Therefore,  if 
we  attempt  the  division  of  any  supposed  prime,  by  all  the  primes  less 


318  PROPERTIES   OP   NUMBERS.  [ART.  XXII 

than  its  square  root,  and  discover  no  factor,  the  number  is  itself  a 
prime. 

TO    TIND    ALL    THE    DIVISORS    OF   A    NUMBER. 

What  numbers  will  divide  5940  without  a  remainder  ? 
We  first  resolve  the  number  into  all  its  prime  factors, 


by  commencing  with  2  and  dividing  as  often  as  possible, 
by  each  of  the  prime  numbers  in  succession.  We  thus 
find  that  5940  =  22x33X  5  X  H,  or  2  x  2  X  3  X  3  X 


5940 
2970 
1485 
495 


It  may,  therefore,  have  as  many  compos-  ^  ^ 
ite  divisors  as  we  can  form  distinct  products  of  these  -Q  ^ 
prime  factors.  In  order  to  determine  all  the  possible  1 

products,  we  arrange  1,  with  the  powers  of  the  factor 
that  is  employed  the  greatest  number  of  times,  in  a  horizontal 
line.  We  then  multiply  each  of  the  numbers  in  the  first  line,  by 
each  of  the  powers  of  another  factor ;  each  of  the  numbers  of 
the  preceding  lines,  by  each  of  the  powers  of  a  third  factor,  &c., 
as  in  the  following  table. 

1  3  9  27  =  33 


2 
4 

6 
12 

18 
36 

54 
108 

= 

coco  cocbcb  cococococbco 

X 
X 
X 
X 
X 
X 
X 
X 
X 
X 
X 

2 
2* 
5 

2 
22 
11 
2 
22 
5 
2 

22 

X 
X 

X 
X 
X 
X 
X 

5 
5 

11 
11 
11 
5X 
5  X 

11 
11 

5 
10 
20 

15 

30 

60 

45 

90 
180 

135 
270 
540 

= 

11 
22 
44 
55 
110 
220 

33 
66 
132 
165 
330 
660 

99 
198 
396 
495 
990 
1980 

297 
594 
1188 
1485 
2970 
5940 

— 

The  numbers  of  the  first  line  having  been  arranged  as  directed, 
we  multiply  them  separately  by  2  and  22 . 

All  the  numbers  of  these  three  lines,  are  multiplied  by  5,  which 
gives  us  three  new  lines  of  divisors. 

All  the  numbers  of  these  six  lines  are  multiplied  by  11,  which 
gives  us  six  new  lines  of  divisors.  We  thus  obtain  48  numbers 
that  will  divide  5940  without  a  remainder,  and  an  examination 
of  the  table  will  show  that  these  are  all  the  divisors,  since  the 
prime  factors  are  combined  in  every  possible  way. 

We  are  able  to  determine  without  actual  trial,  the  number  of 
exact  divisors  of  any  given  number.  By  the  foregoing  table  we 


§101.]  PRIME   AND   COMPOSITE   NUMBERS.  319 

perceive  that  33  had  4,  or  3  +  1  divisors.  33  X  22  has  12,  or 
JT+T  x"^1-  33  X  22  x  5  has  24  or  3  +  1  x  2  +  1  x  1  +  1. 
In  like  manner  each  new  factor  can  be  multiplied  Iby  all  the 
preceding  divisors,  as  many  times  as  are  equivalent  to  the  expo 
nent  of  its  power,  thus  forming  so  many  new  divisors  to  be  added 
to  the  preceding.  Hence,  for  finding  the  number  of  divisors  of  any 
given  number,  we  have  the  following 

RULE. 

Add  1  to  the  exponent  of  each  of  the  prime  factors  of  the  given 
number,  and  multiply  together  the  exponents  thus  increased.  The 
product  thus  obtained,  is  the  number  of  divisors  sought. 

If  any  other  number  than  10  were  adopted  as  the  base  of  a 
system  of  numeration,  the  number  preceding  the  base  would  have 
the  same  properties  as  the  figure  9  in  our  present  system.  For 
example,  1183,  expressed  by  a  scale  of  8  would  be  2237.  The 
sum  of  the  digits  2  +  2  +  3  +  7  =  14  being  divisible  by  7,  the 
number  itself  is  so  divisible. 

A  perfect  number  is  one  that  is  equal  to  the  sum  of  all  its  aliquot 
parts.  Thus,  6,  the  aliquot  parts  of  which  are  1,  2,  and  3,  is  a 
perfect  number,  because  1+2  +  3=6.  The  following  are  the  only 
perfect  numbers  known : 

6,  28,  496,  8128,  33550336,  8589869056,  137438691328, 
2305843008139952128,  2417851639228158837784576, 
9903520314282971830448816128.  Every  perfect  number  must  ter 
minate  either  in  6  or  in  28.  « 

Two  numbers  are  said  to  be  amicable,  when  each  is  equivalent 
to  the  sum  of  all  the  aliquot  parts  of  the  other.  Thus,  220  and 
284  are  amicable  numbers,  because  220  =  1  +  2+  4+  71  +  142, 
which  are  the  aliquot  parts  of  284,  and  284  =1+2  +  4  +  5  + 
10+  11  +  20  +  22  +  44  +  55  +  110,  which  are  the  aliquot  parts 
of  220.  There  are  very  few  amicable  numbers  known. 

EXAMPLES. 

1.  Which  of  the  following  numbers  are  prime  ?     733 ;  949 ;  917 ; 
619;   1009;   1001;   989;   11571. 

2.  Find  all  the  prime  factors  of  780  ;  468  ;  3944  ;  6972 ;  1849; 
2899;  883;  15664. 

3.  Find  all  the  divisors  of  94800;  21100;  6922. 


320  PROPERTIES   OF   NUMBERS.          [ART.  XXII. 

4.  How  many   divisors   has   20736?    44100?   29930?    5940? 
16384?  15309? 

5.  Show  that  if  7  were  adopted  as  the  base  of  a  numerical 
scale,  548712  would  be  expressed  in  such  a  manner  that  the  sum 
of  its  digits  would  be  divisible  by  6. 

6.  Prove  that  137438691328  is  a  perfect  number. 

7.  Prove  that  17296  and  18416  are  amicable  numbers. 

8.  Prove  that  9363584  and  9437056  are  amicable  numbers. 

9.  Show  that  120  and  672  are  each  equal  to  half  the  sum  of 
their  aliquot  parts. 

IO3.     FIGURATE  NUMBERS. 

Figurate,  or  polygonal  numbers,  are  formed  by  adding  the  suc 
cessive  terms  of  an  arithmetical  series. 

Thus,  if  we  add  the  successive  terms  of  the  natu 
ral  series,  1,2,   3,  4,   5,  6,   7,  &c.,  we  obtain  the  ' 
figurate  series  1,  3,  6,  10,  15,  21,  28,  &c.,  which  are  .    *    '    " 

called  triangular  numbers,  because  they  can  always         .    . 
be    arranged   in   the   form   of   an    equilateral   tri 
angle. 

If  we  take  the  arithmetical  progression  1,  3,  5,  7,  *    * 

9,  &c.,  in  which  the  common  difference  is  2,  we  ob-       •  *    * 

tain  the  figurate  series,  1,  4,  9,  16,  25,  &c.,  which       •    •    • 
are  called  square  numbers,  because  they  can  always 
be  arranged  in  the  form  of  a  square. 

The  arithmetical  progression  1,  4,  7,  10,  &c.,  in  which  the 
common  difference  is  3,  furnishes  the  figurate  series  1,  5,  12,  22, 
&c.,  which  are  called  pentagonal  numbers,  because  they  can  always 
be  arranged  in  the  form  of  a  polygon  with  five  sides. 

In  a  similar  manner  are  produced  the  hexagonal,  heptagonal, 
octagonal,  and  other  figurate  series,  the  number  of  sides  of  the 
polygon  in  which  the  numbers  can  be  arranged,  being  always 
two  greater  than  the  common  difference  of  the  arithmetical  pro 
gression  from  which  they  are  derived. 

EXAMPLES. 

1.  Find  the  first  20  triangular  numbers. 

2.  What  are  the  first  20  pentagonal  numbers  ? 

3.  What  are  the  first  10  hexagonal  numbers  ? 


§103.]  THE   FUNDAMENTAL  RULES.  321 

4.  Find  the  first  10  dodecagonal  numbers. 

5.  Find  the  first  five  17-gonal  numbers. 

1O3.    THE  FUNDAMENTAL  RULES. 

It  may  readily  be  perceived  that  the  rules  of  Arithmetic  merely 
indicate  convenient  modes  of  obtaining  a  desired  result,  and  that, 
in  many  cases,  a  variety  of  processes  will  suggest  themselves, 
either  one  of  which  will  serve  our  purpose. 

Bat  it  may  at  first  seem  incredible,  that  the  sum  of  any  number 
of  quantities  can  be  obtained  without  addition ;  the  difference  of 
two  numbers,  without  subtraction ;  the  product  of  two  numbers, 
without  multiplication ;  and  the  quotient  of  two  numbers,  without 
division.  Yet  such  is  the  case,  and  the  following  rules  are  not 
only  interesting  from  their  curiosity,  but  from  the  connexion 
which  they  show  between  the  several  operations  that  may  be  per 
formed  upon  numbers. 

I.   To  obtain  the  sum  of  a  series  of  numbers,  by  subtraction. 
Assume  any  number  larger  than  the  required  sum,  and  from 
the  assumed  number  subtract  in  succession  each  of  the  given 
numbers.     Subtract  the  final  remainder  from  the  number  first 
assumed,  and  the  result  will  be  the  sum  required. 

EXAMPLE. — Find  the  sum  of  69,  93,  and  237. 
Assume  1000        1000 

69  G01  PROOF. 


931    399  Ans.   1000— (1000  — 69  — 93— 237)  = 
93  69  +  93  +  237. 

83~8 
237 

601 

II.  To  find  the  difference  of  two  numbers,  by  multiplication  and  division. 
Write  nine  times  the  subtrahend  under  the  minuend,  and  add 
each  figure  of  the  upper  number  to  the  figures  in  the  same  place 
and  all  the  inferior  places  of  the  lower  number,  carrying  as  in 
ordinary  addition.  Proceed  in  this  manner,  stopping  at  the  figure 
that  falls  immediately  under  the  left  hand  figure  of  the  minuend, 
and  the  result  will  be  the  difference  sought. 

EXAMPLE. — Required  the  difference  between  874  and  10767. 
21 


322  PROPERTIES   OP   NUMBERS.  [ART.  XXII. 


=  13;  1  to  carry  -f  5  +  6  +  6=18;     10757 
1  to  carry  +  7  +  8  +  6  +  6  =  28;  2  to  carry       7866  =  9  x  874 
+  0  +  7  +  8  +  6  +  6  =  29;  2tocarry  +  l+      09883  Am. 
7  +  8  +  6  +  6  =  30,  but  as  the  0  falls  under 
the  left  hand  figure  of  the  minuend,  \ve  stop  there,  and  find  the 
true  remainder  to  be  9883. 

III.  To  find  the  product  of  two  numbers  by  addition,  subtraction  and 

division. 

Resolve  either  factor  into  a  number  of  submultiples  of  some 
power  of  ten,  which  submultiples,  when  combined,  either  by 
addition  or  subtraction,  will  reproduce  the  original  factor. 

Write  1  as  a  common  numerator,  and  the  submultiples  as  de 
nominators  of  a  series  of  fractions,  and  divide  the  other  factor 
by  the  decimal  expression  for  each  of  the  fractions  thus  formed. 
The  several  quotients,  combined  in  the  same  manner  as  the  original 
submultiples,  will  give  the  product  desired. 

EXAMPLE.     What  is  the  product  of  57G9  X  2841  ? 
5769  =  5000  +  500  +  250  +  20  —  1. 

sJoo  +  5k  +  zlv  +  sV  -  }  =  .0002+.002+.004  +  .05-1 
Mte  +  SftV  +  ?&Y  +  2$i--~i-  =  16389729  An,. 

IV.  To  find  the  quotient  of  two  numbers,  by  addition  and  multiplication. 

1.  Employ  as  a  multiplier,  the  difference  between  the  given 
divisor,  and  the  least  power  of  10  which  is  larger  than  the  divisor. 

2.  Write  the  first  figure  of  the  dividend  in  the  quotient,  and 
add  the  product  of  the  first  dividend  figure  by  the  employed  mul 
tiplier,  to  as  many  of  the  succeeding  figures  as  are  equivalent  to 
the  number  of  figures  in  the  divisor.     If  the  sum  has  a  greater 
number  of  figures  than  the  divisor,  write  the  left  hand  figure  under 
the  figure  in  the  quotient,  and  proceed  as  before,  until  a  sum  is 
obtained  having  the  same  number  of  figures  as  the  divisor. 

3.  To  this  sum  annex  the  remaining  figures  of  the  dividend. 
Place  the  left  hand  figure  of  the  result  as  the  second  quotient 
figure,  and  proceed  as  in  paragraph  (2).     Continue  this  process 
until  all  the  figures  of  the  dividend  have  been  employed. 

4.  Add  the  several  figures  in  the  quotient,  and  cut  off  from  the 
right  hand  as  many  figures  as  there   are  in  the  divisor.     The 
figures  so  cut  off  will   represent  a  remainder  ;    the   remaining 
figures  are  the  quotient. 


TM 


\ 


§103.]  THE   FUNDAMENTAL   RUfc^S.  323 

^^^~ii7>,'  '    ' 

[If  the  remainder  is  larger  than  the  divisor,  subtract  the 
divisor  as  often  as  possible,  and  increase  the  quotient  by  a 
number  equivalent  to  the  number  of  subtractions. 

If  the  divisor  is  contained  two  or  more  times  in  the  next  larger 
power  of  10,  the  quotient  may  be  obtained  more  readily,  by 
employing  as  a  multiplier  the  difference  between  some  power  of 
10  and  the  greatest  multiple  of  the  divisor  contained  in  it.  The 
quotient  thus  found,  should  be  multipled  in  the  same  manner  as 
the  original  divisor.] 

EXAMPLE  1.     Required  the  quotient  of  284175  by  89. 
100  —  89  =  11,  the  employed  multiplier. 
284175(2182 
11 

"3192 


Ans.  8192fJ. 


The  several  multiplications  and  additions  may  be  made  men 
tally,  and  the  quotient  thus  obtained  by  employing  very  few 
figures.  If  the  divisor  is  very  near  some  power  of  ten,  or  a  sub- 
multiple  of  any  number  which  is  but  little  smaller  than  some 
power  of  ten,  the  quotient  can  be  obtained  in  this  way  with  great 
facility. 

EXAMPLE  2.     Divide  573612  by  9.  57361  2 

53634.6 
1  1 

Ans.        63734| 

EXAMPLE  3.     Divide  47281591  by  997.  47281  591 

Ans.  47423||fi.  47423.860 

When  the  divisor  consists  entirely  of  9's,  the  following  rule  for 
obtaining  the  quotient  will  be  found  more  convenient : 


324  PROPERTIES   OF   NUMBERS.          [ART.  XXII. 


RULE  FOR  DIVIDING  BY  9's. 

"When  the  divisor  consists  of  any  number  of  9's,  increase  it  by 
1,  for  a  new  divisor.  Divide  the  dividend  by  this  new  divisor. 
By  the  same  divisor,  divide  the  integers  of  the  quotient,  and  pro 
ceed  in  a  similar  manner,  until  a  quotient  is  obtained  less  than 
the  divisor.  Add  all  the  quotients  together,  observing  the  number 
of  units  carried  from  decimals  to  integers.  Add  this  number  to 
the  right  hand  decimal  figure,  and  the  integers  will  represent  the 
quotient,  and  the  decimals  the  remainder.  When  all  but  the 
units  figure  of  the  divisor  are  9's,  increase  the  divisor  by  the 
difference  between  the  units  figure  and  10,  and  divide  as  above 
directed,  multiplying  each  quotient  after  the  first,  by  the  number 
added  to  the  divisor.  Multiply  the  number  carried  from  decimals 
to  integers,  by  the  number  added  to  the  divisor,  and  add  the  pro 
duct  to  the  decimals  for  the  true  remainder.  If  this  increased 
remainder  exceeds  the  divisor,  increase  the  quotient  by  1,  and 
subtract  the  divisor  from  the  remainder  for  the  true  remainder. 

EXAMPLES  FOB.  ILLUSTRATION. 

Divide    8905473    by   999.      The   divisor, 

increased  by  1,  is  1000.     Dividing  by  the  3905.473 

rule,   and  adding  the  quotients,  we   obtain  8.905 

8914.386.  There    being    1    unit    to    carry  8 
from  decimals,  we  add  1  to  the  right-hand  3914  336 
decimal   figure,    and    find    the   quotient    is  l 

8914.387.  8914.387  quotient. 

This  rule  is  founded  on  the  decimal  value  of  -gij^j  and  it  will 
be  easily  seen  that  the  process  is  nearly  the  same  as  in  the  mul 
tiplication  by  .OOlOOi  -f-.  In  dividing  the  numerators  of  frac 
tions  obtained  by  the  multiplication  of  circulating  decimals,  the 
rule  will  often  be  of  use. 

Divide  1549638144  by  9991.     The 
divisor,  increased  by  9,  is  10000.     Di-       154963.8144 

-I    OQ      X££*'7 

viding  first  by  this  number,  we  multi- 
ply  the  integers  of  the  first  quotient 


by  9,  writing  the  first  figure  of  the       155103.4062 
product  under  the  right  hand  decimal 

figure,  which  is  equivalent  to  multi-  .4071  remainder. 

plying  by  9,  and  dividing  by  10000. 


§104.]  CURIOUS   PROBLEMS.  325 

We  multiply  the  integers  of  this  second  number,  and  write  them 
in  the  same  manner,  and  add  the  several  numbers  together. 
There  being  1  unit  to  carry  from  decimals,  we  multiply  it  by  9, 
and  add  the  product  to  4062,  which  gives  4071  for  the  true 
remainder. 

The  foregoing  rules  furnish  us  with  a  general  method  for  deter 
mining  whether  any  number  is  divisible  by  any  other  given  num 
ber,  without  actually  performing  the  division. 

EXAMPLES. 

1.  Find  by  subtraction,  the  amount  of  1690,  84,  207,  168,  and 
4493. 

2.  Find  by  multiplication  and  addition,  the  difference  between 
847  and  10082  ;  between  19804  and  2973. 

3.  Multiply  764  by  1972,  by  Case  III. 

4.  Divide  21709  by  837,  by  Case  IV. 

5.  Divide  684291708  by  9  ;  by  97 ;  by  99 ;  by  995  ;  by  990; 
by  999 ;  by  9999. 

1O4.     CURIOUS  PROBLEMS. 

1 .  To  add  a  column  of  numbers  at  a  glance. 

The  numbers  to  be  added  should  be  arranged  in  pairs,  each 
member  of  the  pair  being  the  arithmetical  complement  of  the 
other.  The  key  line  may  be  written  in  the  middle,  and  the  sum 
of  the  whole  may  be  found  by  prefixing  the  figure  which  repre 
sents  the  number  of  pairs,  to  the  key  line.  For  example,  if  we 
first  write  the  three  numbers,  4719082,  3604227,  4719082 
1518729,  and  reserve  the  third  as  a  key  line,  then  3604227 
take  each  figure  of  the  second  number  from  9  except  1518729 
the  right  hand  figure,  which  we  take  from  10,  we  6395773 
obtain  a  fourth  number,  6395773.  Proceeding  in  a  5280918 
similar  manner  with  the  first  number,  we  obtain  21518729 
5280918  for  our  fifth  number,  thus  completing  two 
pairs  besides  the  key  line.  Then  prefixing  2  to  the  key  line,  we 
obtain  21518729  for  the  sum  of  the  five  numbers. 

It  will  be  more  difficult  to  detect  the  method  pursued,  if  we 
subtract  each  figure  of  one  of  the  numbers  from  8,  except  the 
right  hand  figure,  which  we  take  from  9.  Each  figure  of  the  key 
line  must  then  be  diminished  by  1 .  Other  variations  may  be  made 


326  PROPERTIES   OP  NUMBERS.          [ART.  XXII. 

in  the  process,  and  the  position  of  the  key  line  in  the  column  may 
be  altered  at  pleasure. 

2.  To  tell  two  or  more  numbers  which  a  person  has  thought  of,  neither 

number  being  greater  than  9. 

Direct  the  person  to  double  the  first  number  thought  of,  add  1 
to  the  product,  multiply  the  sum  by  5,  and  add  to  the  product  the 
second  number.  If  there  be  a  third,  let  him  double  the  first  sum, 
and  add  1  to  it,  multiply  by  5,  and  add  the  third  number.  Thus 
proceed  for  each  additional  number  thought  of.  Finally,  if  there 
were  but  two  numbers  thought  of,  direct  him  to  subtract  5  from 
the  result;  if  three,  55  ;  if  four,  555,  and  so  on.  The  remainder 
will  be  composed  of  the  figures  thought  of,  in  their  proper  order. 

For  example,  suppose  the  numbers  thought  of  to  be  2,  2,  8,  9. 
2x2+1=5;  5x5  =  25;  25  +  2=27;  27x2  +  l=55;55 
X5  =  275;  275+8=283;  283x2  +  1=567;  567x&  = 
2835;  2835  +  9  =  2844;  2844  —  555=2289,  the  figures  of 
which  indicate,  in  their  order,  the  four  numbers  thought  of. 

3.  What  is  the  product  of  £11  11s.  lid.  by  £11  11s.  lld.1 
Questions  similar  to  this  are  to  be  found  in  many  of  the  old 
arithmetics,  and  different  answers  have  been  given,  according  to 
the  different  views  of  the  proposers.  But  in  reality  the  problem 
is  absurd,  the  error  consisting  in  the  supposition  that  applicate, 
or  concrete  numbers,  are  capable  of  being  multiplied  together. 
The  thorough  arithmetician  should  never  lose  sight  of  the  fact, 
that  all  the  operations  of  arithmetic  are  performed  on  abstract 
numbers.  We  say,  indeed,  that  the  area  of  any  rectangular  sur 
face  is  found  by  multiplying  the  length  by  the  breadth ;  but  this 
is  merely  a  convenient  expression,  adopted  to  avoid  circumlocu 
tion.  Our  meaning  is,  that  if  we  multiply  the  NUMBER  of  feet  in 
the  length,  by  the  NUMBER  of  feet  in  the  breadth,  the  product 
will  represent  the  NUMBER  of  square  feet  in  the  area.  So  in 
geometry,  when  we  say  AB  x  CD  =  the  rectangle  AD,  we  mean 
that  if  the  line  AB  were  repeated  for  every  point  of  the  line  CD, 
we  should  have  the  surface  AD. 

If  it  were  possible  to  form  a  product  of  concrete  numbers,  we 
should  be  obliged  to  call  the  product  of  1£  by  1£,  1  square 
found;  Is.  X  ls->  would  then  be  1  square  shilling ;  Id.  X  Id.  =  1 
square  d.  ;  Iqr.  X  lqr->  =  1  square  qr.  1  sq.  £  would  then  be  equal 
to  400  sq.  s.  ;  I  sq.  s.  =  144  sq.  d.  ;  Isq.  d.  =  16  sq.  qr.  ;  and  the 
product  of  £11  11s.  lid.  by  £11  11s.  lid.,  would  be  134  sq.  £ 


§  105.]  CHRONOLOGY.  327 

185  sq.  s.  49  sq.  d.  The  impossibility  of  conceiving  of  a  square 
pound,  a  square  shilling,  or  a  square  penny,  shows  at  once  the 
absurdity  of  the  original  question.  But  the  investigation  is  a 
useful  one,  both  because  it  furnishes  exercise  in  an  intricate  kind 
of  multiplication,  and  because  it  shows  the  error  of  the  very 
prevalent  idea,  that  dollars,  cents,  and  mills,  can  be  multiplied  by 
dollars,  cents,  and  mills. 

The  number  of  curious  problems  might  be  extended  indefi 
nitely,  if  our  limits  would  allow.  The  few  here  given  may  suffice 
to  excite  an  interest  in  such  investigations,  and  lead  the  pupil  tc 
exercise  his  own  ingenuity,  and  to  consult  the  works  of  authors 
who  have  treated  the  subject  more  fully. 


XXIII.    MISCELLANEOUS  PROBLEMS. 

1  O*>.     CHRONOLOGY. 

ACCORDING  to  the  Julian  Calendar  or  OLD  STYLE,  the 
solar  year  was  considered  as  being  365  days  and  6  hours. 
The  6  hours  in  4  years  amounted  to  a  day,  therefore  every 
fourth  year  was  called  a  Leap  Year,  and  consisted  of  366 
days. 

But  the  true  solar  year  is  about  11  minutes  less  than  the 
Julian  year,  and  on  this  account,  in  1582,  it  was  found 
that  spring  commenced  10  days  later  than  at  the  estab 
lishment  of  the  Julian  Calendar.  Pope  Gregory  XIII., 
therefore,  caused  ten  days  to  be  taken  out  of  the  month  of 
October  in  that  year,  and  to  prevent  the  recurrence  of  a 
similar  variation,  he  ordered  that  the  centurial  years  should 
not  be  regarded  as  leap  years,  unless  the  number  of  cen 
turies  were  divisible  by  4. 

This  computation,  which  is  called  the  Gregorian  or  NEW 
STYLE,  was  soon  adopted  in  the  greater  part  of  Europe ; 
but  in  England  and  America,  the  change  was  not  made 
until  1752,  when  the  error  had  amounted  tc  eleven  days. 


328  MISCELLANEOUS   PROBLEMS.      [ART.  XXIII. 

It  was  then  ordered  that  the  3d  of  September  should  be 
called  the  14th,  and  the  Gregorian  calendar  adopted  for  the 
future.  In  Russia,  the  Old  Style  was  retained  until  the 
year  1830. 

One  of  the  first  seven  letters,  A,  B,  C,  D,  E,  F,  Gr,  is 
attached  to  every  day  in  the  year ;  thus  A  is  applied  to  Jan. 
1st,  8th,  15th,  &c. ;  B,  to  Jan.  2d,  9th,  16th,  &c. ;  C,  to 
Jan.  3d,  10th,  17th,  &c.  In  this  manner  all  days  in  any 
year  which  have  the  same  letter,  fall  on  the  same  day  of 
the  week.  The  DOMINICAL  LETTER  for  any  year  is  the  letter 
that  falls  against  all  the  Sundays.  Thus,  the  6th  of  Janu 
ary,  1850,  fell  on  Sunday,  and  the  dominical  letter  was, 
therefore,  the  6th  letter,  or  F.  But  in  leap  year  there  are 
two  dominical  letters,  the  first  for  January  and  February, 
the  second  for  the  remainder  of  the  year. 

PROBLEMS. 

I.    To  find  the  dominical  letter  for  any  year,  according 

to  the  Julian  or  OLD  STYLE. 

To  the  given  year  add  one  fourth  of  itself,  plus  4,  and  divide 
the  sum  by  7.  If  there  is  no  remainder,  the  dominical  letter 
is  G;  if  1  remainder,  F;  and  so  on  in  inverse  order.  If 
the  given  year  be  leap  year,  the  letter  thus  found  will  be 
the  dominical  letter  for  the  last  10  months,  and  the  next 
following  letter,  for  the  remainder  of  the  year. 
What  was  the  dominical  letter  for  A.  D.  1531  ? 

To  the  given  year,  we  add  Given  Jear     1531 

one  fourth  of  itself,  (rejecting  one-fourth      382 

the  fraction,)  and  4.  Dividing  

this  sum  by  7,  we  have  a  re-  *              7  )  1917 

mainder  6,   which    indicates  273 +6  remainder. 

that     the     dominical     letter 

sought  is  the  6th  from  G,  counting  in  retrograde  order,  which 

is  A. 

What  were  the  dominical  letters  for  A.  D.  564  ? 


§  105.]  CHRONOLOGY.  329 

The  remainder  2  indicates  that  the  dominical  564 
letter  is  the  2d  from  G,  or  E.  But  the  year  being  141 
leap  year,  the  dominical  letter  for  January  and 

February,  will  be  the  next  following,  or  F.     The  7  )  709 

two  letters  sought  are  therefore  F,  E.  "~ToT_f_  2 

If  the  given  year  were  before  the  Christian  era,  the 
remainder  would  indicate  the  direct  order  of  the  letters. 
Thus,  1  denotes  A;  2  denotes  B;  5,  E,  &c. 

II.  To  find  the  dominical  letter  for  any  year,  according 

to  the  Gregorian  or  NEW  STYLE. 

Divide  the  centuries  by  4,  and  take  the  remainder  from 
3.  Add  twice  this  remainder  to  f  of  the  odd  years,  and 
divide  the  sum  by  7.  If  there  is  no  remainder,  the  domi 
nical  letter  is  Gr ;  if  1  remainder,  F,  &c.,  as  in  the  preceding 
rule. 

What  is  the  dominical  letter  for  1895  ? 

4 )  18  cent. 

Dividing  18  centuries  by  4,  the  remainder  7~T"  9 

is  2.     Taking   this   remainder    from    3,   we  3 2=1 

have  a  remainder  of  1.     Twice  1  added  to  2x1  =  2 
95  years  plus  J  of  95,  (rejecting  the  fraction,)  0(ld  Jears  95 
gives  120,  which,  divided  by  7,  gives  a  re 
mainder    1,    indicating    that   the    dominical  7)120 
letter  is  the  1st  before  G,  which  is  F.  JJ7    •   j 

III.  To  find,  the  day  of  the  weeTc  corresponding  to  any 

given  day  of  the  month. 

The  dominical  letter  found  by  one  of  the  preceding 
rules,  will  indicate  the  day  on  which  the  first  Sunday  in 
January  will  fall.  The  day  of  the  week  for  the  correspond 
ing  day  of  each  succeeding  month,  may  be  found  by  the 
initials  of  the  following  couplet : — 

At  Dover  Dwell  George  Brown,  Esquire, 
Good  Captain  French,  And  David  Friar. 

On  what  day  of  the  week  was  the  Declaration  of  Independence 
signed  ? 


330  MISCELLANEOUS    PROBLEMS.        [ART.  XXIII. 

The  dominical  letters  for  1776  were  G,  F.  Therefore,  the  first 
Sunday  in  January  was  the  7th  of  the  month.  Then  A  repre 
senting  the  7th  Jan.,  D  would  represent  the  7th  Feb. ;  I)  the  7th 
March ;  G  the  7th  April ;  B  the  7th  May ;  E  the  7th  June,  and  G 
the  7th  July.  But  1776  being  a  Leap  Year,  the  dominical  letter 
after  February  is  one  day  earlier  in  the  month,  and  a  day  of  the 
month  which  would  otherwise  be  represented  by  G,  will  be  repre 
sented  by  A  or  Sunday.  The  7th  July,  therefore,  came  on  Sunday, 
and  the  4th  on  Thursday. 

The  initials  0.  S.  denote  the  Old  Style.  In  all  cases  not 
thus  marked,  the  New  Style  is  understood. 

EXAMPLES. 

1.  Washington  was  born  on  the  22d  Feb.  1732.     What 
was  the  day  of  the  week  ?  Ans.  Friday. 

2.  The  pilgrims  landed  at  Plymouth,  Dec.  11,  1620.  0. 
S.     What  was  the  day  of  the  week  ?          Ans.  Monday. 

3.  The  Battle  of  Waterloo  was  fought  June  18,  1815. 
Is  it  probable  that  a  letter,  purporting  to  have  been  written 
at  the  time,  and  dated  Friday,  June  18,  is  authentic  ? 

Ans.  No;  because  the  battle  was  fought  on  Sunday. 

4.  Constantine  Paleologus  was  the  last  Christian  Emperor 
of  Constantinople.     On  the  29th  of  May,  1453,  the  city 
was  taken  by  the  Turks,  the  Emperor  Constantine  killed, 
and  Mohammed  II.  ascended  the  throne,  thus  founding  the 
present  empire  of  Turkey  in  Europe.     What  was  the  day 
of  the  week  ?  Ans.  Tuesday. 

CONTEMPORARY  PRINCES, 

FROM   EGBERT    TO    QUEEN   VICTORIA. 

The  following  table  is  compiled  principally  from  Wade's  British 
History.  The  teacher  will  find  in  it  the  materials  for  a  great  va 
riety  of  Chronological  questions. 

In  order  to  determine  the  date  of  the  commencement  of  each 
reign,  add  the  number  following  the  name  to  the  number  at  the 
head  of  the  column.  Ex. :  Ethelwolf,  836 ;  Athelstan,  925. 


§105.] 


CHRONOLOGY. 


331 


800. 

900. 

1000. 

ENGLAND.  —  Egbert;    E- 

ENG.  —  Edward   the  El 

ENG.  —  Edmund  Ironside, 

thelwolf,    36  ;     Ethel- 

der,  1;  Athelstan,  25; 

16;    Canute  the  Great. 

bald,  57;  Ethelbert,60; 

Edmund,    41  ;    Ed  red, 

17;    Harold   Ilarefoot, 

Ethelred  I.,  66;  Alfred 

46;  Edvvy,  55;  Edgar, 

36;    Hardicanute,  39  ; 

the  Great,  72. 

59;  Edw.  the  Martyr. 

Edward  the  Confessor, 

75;  Ethelred  II.,  78. 

41  ;    Harold    II.,    65  ; 

Wrn.  I.,  the  Conquer 

or,  66;  Wm  II.Rufus, 

87. 

SCOTLAND.  —  Achaius. 

SCOT.—  Constant.  III.,  1  ; 

SCOT.—  Malcolm   II.,  4; 

787;  Congale  III..  19; 

Malcolm  I  ,38;  Indul- 

Duncan,  34  ;  Macbeth, 

Alpin,3i;  Kenneth  II.. 

phus,  58  ;  Duphus,  68  ; 

40;  Malcolm  III.,  57; 

34  ;    Donald    V.,    54  ; 

Cullenus,  72;  Kenneth 

Donald  VI.,  93;   Dun 

Constantino     II.,    58; 

III.,    73;    Const.    IV., 

can  II.,  94  ;  Edgar,  96. 

Ethus,  74  ;  Gregorv  the 

94  ;  Grimus,  97. 

Great,  75;  Donald"  VI., 

FR.—  Henry  I.,  31;  Phi 

9-2. 

lip  I.,  60. 

FRANCE.—  Charlemagne; 
Louis  I.,   14;    Charles 

FR.—  Robert,  22;  Ralph, 
23;  Louis  IV.  ,36;  Lo- 

GER.—  Henry     II.,     the 
Saint,    2;    Conrad    the 

the    Bald,    43;    Louis 

tharius,  54;  Louis  V., 

Salic,  24:  Henry  III.. 

II.,  the  Stammerer,  77  ; 

86;    Hugh  Capet,   87;!     39;  Henry  IV.,  56. 

Louis  III.  and  Carlo- 

Robert  the  Pious,  97.     SP.—  Sancho    III.,     the 

man,  79;     Charles  the 

Great  ;    Ferdinand   I  , 

Fat,    84  ;    Hugh,    88  ; 
Charles  the  Simple,  98. 

GER.  —  Conrad    I.,    11; 

in  Castile,  33  ;   Garcia 
IV.  in  Navarre,  Rami 

GERMANY.  —  Charle 

Henry  I.,  19;  Otho  the 
Great,   36;    Otho   II., 

rez  I.  in  Arragon,  35; 
Sancho  IV  ,  Nav.,  51  ;  : 

magne;  Louis   I.,    14; 

73;  Otho  III.  .83. 

Sancho    I.,    Arr.,    63; 

Louis    II.,  43;    Carlo- 

Sancho   I.,   Cast.,    65; 

man,    Louis    III.,   the 

Alfonzo   I.,  Cast.,  72; 

Younger,  and  Charles 

SP.—  Sancho  I.,  5;  Gar- 

Sancho  V  ,    Nav.   and 

the    Fat,    76;    Arnold, 

ci;i  II.  ,26;  Sancho  11., 

Arr.,  76;  Peter  I.,  N. 

87;  Louis  IV.,  the  In 

70;  Garcia  III.,  94. 

and  A.,  94. 

fant,  99. 

PA.  ST.—  Romanus  For- 

PA.  ST.—  Jno.  XVII.  find 

SPAIN.  —  Garcia    I.,    58  ; 

mosus  and  John  IX.,  1  ; 

XVIII.,3;SenriusIV., 

Fortanio,  80. 

Benedict  IV.,  5;    Leo 

9;  Benedict  VIII-.  12; 

V.  and  Christopher,  6; 

Jno.  XIX.,  21  ;  Bened. 

PAPAL     STATES.  —  Leo 

Sergius  III  ,  7;  Anas- 

IX.,  33;   Gregory  VI., 

III.,  795;  Stephen  V., 

tasius,  10;  Lando  find 

44;    Clement    II..   47; 

16  ;     Paschal    I.,    17  ; 

John   X.,  12;  Leo  VI., 

Dnnvisia   II.,  48;   Leo 

Eugene    II.,   20;    Va 

28;  Stephen  VIII.,  29; 

IX..  49;  Victor  II.  .55; 

lentine,    24  ;     Gregory 

John  XI.  31;  LeoVII., 

Stephen  X,  57;  Nicho 

IV.,   27;    Sergius   II., 

36;    Stephen  IX.,  40; 

las  II.,  58;  Alexander 

43;   Leo  IV.,  47;    Be 
nedict  III.,  55;   Nicho 

Martin    II.,   43;    Aga- 
pet  II.,  46;  John  XII., 

II.,  6!  ;  Gregory  VII., 
73;    Victor    III.,    85; 

las  I.,  58;  Adrian  II., 

56;   .Benedict   V.,   65; 

Urbnn  II.,  87;  Pascal 

66;    John    VIII.,    73; 

John  XIII.  ,66;  Donus 

II.,  99. 

Martin  I.,   83;  Adrian 

II.  and   Benedict  VI., 

Rus.—  Swatopolk  I.,  15; 

III.,  84;  Stephen  VI., 

73;  Benedict  VII.  ,74; 

Jaroslaw  I  ,  of  Kiew, 

85;  Forrnosus,91;  Ste 

John   XIV,   84;  John 

18  ;     Isaslaw    I.,    51  ; 

phen  VII.  ,97. 

XV.,  85;   John  XVI., 

Swatosl  iw     II.,     73  ; 

86;     Gregory    V.,    96; 

Wsewolod  I  ,78;  Swa 

Silvester  I'j.,  99. 

topolk  II.,  93. 

RUSSIA. 

Rus.—  Ighor  I.  13  ;  Swa- 

CON.  —  Const.  IX.,  nlone, 

toslaw   I      45  ;    Jaro- 

25;  RoiriiinusIlL,  28; 

polk  I  ,  72  ;  Waldimer 

Michael    IV.,  34;   Mi 

the  Great,  80. 

chael  V.,  36;   Zoe  and 

CONSTANTINOPLE. 

CON.  —  Alexander,    11  ; 

Theodora,   and  Const. 

—Irene,  797;  Nicepho- 

Rom-mus,  19;  Constan 

X.,  42;  Theodora,  54; 

rus  I.,  2;  Michael  I., 

tino  VII.  ,45;  Romanus 

Mich.  VI.,  56:    Isaac 

11;    Leo  V.,    13;    Mi 

II.  ,59;  NicephorusII.,      I.,  57;  Const    XI,,  59; 

chael  II,  20;  Theophi- 

63;  John  Zimisces,  69;;     Eudocia  and  Ronnnus 

lus,  29;  Michael   III., 

Basil  II.  and  Con.VIlI., 

III.,   67;    Mich.  VII  . 

42;  Basil,  67;  Leo  VI., 

76. 

71;NicephorusIlI.,78; 

86. 

Alexius  I.  ,81. 

332 


MISCELLANEOUS   PROBLEMS.      [ART.  XXIH. 


1100. 

1200. 

1300. 

ENGLAND.  —  Henry    I.  ; 
Stephen,  36;  Henry  II. 

ENG.  —  Henry    III.,    16; 
Edward  I.,  72. 

ENG.  —  Edward    II.,    7; 
Edw.IJI.,27;  Richard 

54;  Richard  I.,   Coeui 

II.,  77;  Henry  IV.,  99. 

de  Lion,  89  ;   John,  99 

SCOTLAND.  —  Alexander 

SCOT.—  AlexanderII.,14; 

SCOT.—  Robert  Bruce,  6  ; 

I.,   7  ;    David    I.,   24 

Alex.  III.,  45;  Marga 

David  II.,  29;  Edward 

Malcolm  IV.  ,53;  Wil 

ret,   86;    John   Baliol, 

Baliol,  32;  Robert  11., 

liam  I.,  65. 

88;  Interregnum,  96. 

70;  Robert  III.  ,90. 

FRANCE.  —  Louis  VI.,  the 

FR.  —  Louis  "VIII.,    23; 

FR.—  LouisX.,K'.  of  Na 

Gross.  8;  Louis  VII. 

Louis  IX..  (St.  Louis.) 

varre,   14  ;    Philip   the 

37;  Philip  II.,  77;  Au 

26;     Philip    III.,    the 

Tall,  K.  of  Nav.,   16; 

gustus,  80. 

Bold,  70;    Philip  IV., 

Chas.   IV.  (the  Fair), 

the  Fair,  85. 

K.  of  Nav.,  22;  Philip 

GERMANY.  —  HenryV.,6 

GER.—  Frederic  II.,   12 

VI.    (the    Fortunate), 

Lotharius  II.,  the  Sax 

Conrad  IV.,  50;  Wm. 

28;  John  I.,  (the  Good,) 

on,    25;    Conrad  III. 

ofHolland,54;  Rich'd, 

50;Chas.V.(theWise), 

38;  Frederick  I.,  Bar- 

Duke  of  Cornwall,  57; 

64;  Chas.  VI.,  80. 

barossa.52;  Henry  VI. 
90;    Philip    and    Otho 

Rodolph  of  Hapsburgh, 
73;  Adolphus  of  Nas 

GER.—  Henry    VII.,    8; 
Louis  of  Bavaria,  and 

IV.,  98. 

sau,  92;  Albert  of  Aus 

Fred,   of  Austria,  14  ; 

tria,  98. 

Chas.   IV.,    46;    Win- 

ceslnus,  78. 

SPAIN.—  Alphonso  I.,  N 

SP.—  James  I.,  Ar.,  13 

SP.—  Alphonso  V.,  C.,  14; 

and  Arr.,  4  ;    Urraca 

Henry  I.,  C.,  14;  Fer-      Alph.  IV.,  Ar.,27;  Jo- 

Gas.,  9;  Alphonso  II. 

din.  III..  C.,  17  ;  Theo 

anna  II.,  N.,28;  peter 

Cas.,   26;    Garcia    V. 

bald  I  ,N.,34;  Alphon 

II.,  Ar.  36;  Chas.  II., 

Nav.,  Ramirez  II.,Ar. 

so  IV,C,52;  Theob.      N.,49;  Peter  I..  C..  50; 

34;  PetronillaandRay- 
mondo,  Arr.,  37;   San 

II.,  N.,  53;  Henry  I., 
N.,  70;  Joanna  I.,  N., 

Henry  II.,  C.,69;  John 
I.,C.,79;  Charles  III., 

cho  VI.,  the  Wise,  N 

74;  Peter  III.,  Ar.,  76; 

N.,86;  JohnI.,A..87; 

50;  Sancho  II.,  Cast., 

Sancho  IV.,  C.84;  Al 

Henry  III.,  C.,  90;  Mar 

57;  Alphonso  II..  Arr., 

phonso   III.,   Ar.,   85; 

tin,  A.,  95. 

62;  S;incho  VII.,   N., 

Jas.  II.,  Ar.,  91;  Ferd. 

94;  Peter  II..  Arr..  96 

IV.,  C.,  95. 

PAPAL    STATES.  —  lelas 

PA.  ST.—  Honorius   III.,  PA.  ST.—  Benedict  X.,  3; 

II.,    18;    Calixtus   II., 

17;    Gregory  IX.,   27;      Clem.V.,  5;  Jno  XXI., 

19;   Honorius   II.,   25; 

Celestine  IV..  41;   In-1     16;  Alex    II,   27;  Be- 

Innocent  11.,   30;    Ce 

noc.  IV.  ,43;  Alex.  IV.,      ned.XI.,34;  Clem  VI., 

lestine  II.,  43;  Lucius 

54;  Urb.IV.,62;  GregJ     42;  Innocent  VI.,   53; 

II.,   44;    Eugene   III., 

X.,  64;   Clem.  IV.,  65;      Urban   V.,   63;    Greg. 

45;  AnastasiusIV.,54; 

Innoc.V.,    Adrian  V.,      XL,  71  ;  Urban  VI.,  78  ; 

Adrian  IV.,  55;  Alex. 

and  John  XX.,  76  ;  Ni-      Boniface  IX.,  90. 

III.,   59;    Lucius   HI., 

cholas  III.,  77;  Martin, 

81  <$     Urb:m    III.,    85; 

IV.  ,81;  Honorius  IV., 

Gregory  VIII.  ,87;  Cle 

85;  Nich.  IV.,  88;  Ce- 

ment  HI.  ,88;  Celestine 

lestineV.,  94  ;  Boniface 

III.,  91;  Innocent  III., 

VIII.,  95. 

RUSSIA.—  Waldimir   II.. 

Rus.  —  Jurje  II.  ,13;  Con- 

RTTS.  —  Michailow,    5; 

13  ;  Mistislaw,  25  ;  Ja- 

stantine,  17;  Jaroslaw 

Jurje  III.  ,17;  IwnnL, 

ropolk   II.,   32;    Wse- 

II.  ,38;  Alexander,  45; 

of  Moscow,  28  ;  Semen, 

wolod  II.,  38;  Isaslaw 

Newskoi.50;  Jaroslaw 

40;  IwanII.,53;  Dimi 

II.,   46;    Jurje   I.,  49; 

III.  ,62;  Wasilejl.,70; 

trej  II.  ,59;  Dimit.IIL, 

Andrej,  57;  Michel  1.. 

Dimitrej,   75;   Andrej. 

63;  Wasilej  II.  ,89. 

75;  WTsewolodIII.,77. 

81;  Danilo,  94. 

CONSTANTINOPLE.  —  Jno. 

CON.  —  Isaac  II.  restored, 

CON.  —  John    Canta,   41  ; 

Comnenus,  18;  Manuel 

3;  Mourzouffe,4  ;  Bald 

John    Paleologus,    55  ; 

Com.  .43;  Alex's  Com., 

win,  4;  Henry,  6;  Pe 

Manuel  Pal.,  91. 

80,     Andronicus,    83; 

ter,    17;    Robert,    19; 

Isaac  II.,  85;  Alexius 

John,  with  Baldwin  II., 

Angelus,  95. 

28;  Baldwin  II.,  alone, 

37;    Michael   Paleolo- 

^us,61;  Andronicus,  82. 

§105.] 


CHRONOLOGY. 


333 


1400. 

1500. 

1600. 

ENGLAND.—  Hen.  V.,  13; 

ENG.—  Henry   VIII.,    9; 

GREAT   BRITAIN.  —  Jas. 

Hen.  VI.,  22;  Edward 

Ed  ward  VI.,  47;  Mary, 

I.,   3;  Charles   I.,   25; 

IV.,  61;  Edw.  V.  and 

53  ;  Elizabeth,  58. 

Cromwell,  53  ;  Charles 

Richard  III.  ,83;  Hen. 

II.,  60;  James  II.,  85; 

VII.,  85. 

Win.  ajid  Mary,  89. 

SCOTLAND.  —  James  I.,  5  ; 

SCOT.  —  James   V.,    12; 

James  II.,  37;    James 

Mary,  42;  James  VI., 

III.  ,60;  James  IV.  ,87 

67. 

FRANCE.—  Charles  VII. 

FR.  —  Francis     I.,     15; 

FR.  —  Louis   XIIL,  10-, 

the  Victor,  22  ;    Louis 

Henry  II.,  47;  Francis 

Louis  XIV.,  the  Great, 

XL,  the   Prudent,  61, 

II.  ,59;  Charles  IX.  ,60; 

43. 

CliarlesVIIL,  the  Affa 

Henry  III.,  74;  Henry 

ble,  83;  Louis  XII.  ,98 

IV.,  the  Great,  89. 

GERMANY.—  Robert;  Si- 

GER.  —  Charles  V  ,    19; 

GER.—  Ma1thias,l2  jFer- 

gismund,  11;  Albert  II., 
37;  Frederic  III.,  40; 

Ferdinand  I.  ,58;  Max 
imilian    II.,    64;    Ro- 

dinand   II.,   19;   Ferd. 
III.,    37;   Leopold    I., 

Maximilian  I.,  93. 

dolph  11.,  76. 

58. 

SPAIN.—  John  II.,  Cast., 

SP.—  Charles  I.  ,16;  Phi 

SP.  —  Philip     IV.,     21; 

6;  Ferdinand  I.,  Arr., 
12;  Alphonso  V.,  Arr., 

lip  II.,  56;  Philip  III., 

Charles  II.,  65. 

16;    Blanche,   N.,   and 

John  I.,  A.,  25;  Henry 

IV..  C.,54;  Ferdin.II. 

and    Isabella,    C.,    74; 

Ferd.  II.,  the  Catholic, 

A.,    79;    Eleanor    and 

Francis  Phoebus,  Nav., 

79;  Catharine,  N.,  83. 

PAPAL   STATES.  —  Inno 

PA.  ST.—  Pius  III.   and 

PA.  ST.—  Leo    XI.    and 

cent  VII.,  4;   Gregory 

Julius  II.,  3;  Leo  X., 

Paul  V.,  5;  Greg.  XV., 

XII.,  6;  Alexander  V., 

13;    Adrian    VI.,    22; 

21;    Urban  VIII.,   23; 

9;    John    XXII.,    10; 

Clement  VIL,  23;  Paul 

Innoc.  X.,  44;  Alexan 

MarH-  xr       7.  EUgene 

III.  ,34;  Julius  III.  ,50; 

der  VII.,   55;  Clement 

IV.,  di;  Nicholas  V., 

Marcellinus     II.,     55; 

IX.,  67;  Clem.  X,  70; 

47;  Calixtus   III.,    55; 

Paul    IV.,    56  ;      Pius 

Innoc.  XL,  76;    Alex. 

Pius  II.,  58;  Paul  II., 

IV.,  59;   Pius  V.,  66; 

VIII.,  89;  Innoc.  XII.. 

64  ;     Sixtus    IV.,    71  ; 

Gregory  XIIL,  72;  Six 

91. 

Innoc.  VIII.  ,84;  Alex. 

tus  V.,  "85;  Urban  VII. 

VI.,  92. 

and  Gregory  XIV.,  90; 

Innocent  IX.,  91;  Cle 

ment  VIII.,  92. 

RUSSIA.  —  Wasilej    III., 

RTTS  .—Wasilej    IV.,    5; 

Rus.  —  Wasilej    Schuis- 

25;  Iwan  Wasilej  III., 

Iwan  Wasilej  IV.,  33; 

koi,6;  Mich.Fedrow- 

62. 

Feodor   I.,   84;    Boris 

itsch,   13;  Alexej,  45; 

G-odunow,  98. 

Feodor  11.,  76  ;    Iwan 

V.,82;  Pet.  the  Great, 

85. 

CONSTANTINOPLE.  —  Jno. 

CON.—  Selim  I.,  12;  So- 

CON.  —  Achmed    I.,    4; 

Paleologus,    25  ;    Con- 
stantine    Paleol.,    48  ; 

lyman,   20;   Selirn   II., 
66  ;  Amurath,  74  ;  Mo 

Mustapha,  17;  Osman 
I.,   18;    Mustapha   re 

Mohammed  II.,  53;  Ba- 

hammed  III.,  95. 

stored.    22  ;    Amurath 

jazet,  81. 

IV.,  23;   Ibrahim,  40; 

Mohammed     IV.,    48; 

Solyman  11.,  87;  Ach 

med  1  1..  91:  Mustapha 

II.,  95. 

834 


MISCELLANEOUS   PROBLEMS.       [ART.  XXIII 


1700. 


1801. 


GKEAT  BRITAIN.— Anne,  2;    George 
I.,  14;  Geo.  II.,  27;   Geo.  III.,  60. 

FR-VXCE. — Louis    XV.,    15;    Louis 
XVI.,  74;  Republic,  92. 


GERMANY.— Joseph  I,  5;  Chas.  VI., 
It;  Chas.  VII.,  42;  Francis  I.  and 

Maria  Theresa,  45  ;  Joseph  II.,  Go  ; 
Leopold  II.,  90;  Francis  II.,  92. 

SPAIN.— Philip  V.;  Ferdinand  VI.. 
51;  Charles  III. ,59;  Chas.  IV. ,68. 

PAPAL  STATES. — Clement  XI.;  Inno 
cent  XIII.,  21 ;  Benedict  XIII.,  24  ; 
Clem.  XII.,  30;  Bened.  XIV.,  40; 
Clem.  XIII.,  58;  Clem.  XIV.,  69; 
Pius  VI.,  75. 

RUSSIA.— Catharine  I.,  25;  Peter  II., 
27  ;  Anne,  30 ;  Iwan  VI.,  40 ;  Eliza 
beth.  41  ;  Peter  III.  and  Catharine 
II.,  62;  Paul  I.,  96. 

CONSTANTINOPLE. — Achmed  III.,  3; 
Mohammed  V.,  30;  Osrnari  II.,  54  ; 
Mustupha  III.,  57;  Abdul-IIamet, 
74;  Selimlll.,  89. 


GT.  BRIT.— George  IV. ,20;  William 
IV.,  30;  Victoria,  37. 

FR. — Napoleon  Emperor,  4;  Louis 
XVIII..  14;  Charles  X.,  24;  Louis 
Philippe,  30  ;  Republic,  48.' 

AUSTRIA.— Francis  I.,  6;  Ferdinand 
I.,  35;  Francis  Joseph  I.,  48. 


Sr.— Ferd.VII.  and  Joseph  Napoleon, 
8;  Ferd.VII.,  14;  Isabella  II.,  33. 

PA.  ST.— Pius  VII.,  Leo  XII.,  23; 
Pius  VIII.,  29;  Gregory  XVI.,  31 ; 
Pius  IX.,  46. 


Rus.— Alexander  I.,  1;  Nicholas  I., 
25. 


CON.— Mohammed  VI.,    8;    Abdul- 
Medjid,  39. 


THE  SMALLER  EUROPEAN  STATES,  FROM  1700. 


1800. 


SWEDEN.— Charles XII. ,1697;  Ulrica  SWE.  —  Charles    XIII.,    9;    Charles 
Eleanora.  19;  Frederic,  20;  Adol-j     John  XIV. ,18;  Oscar  I.,  44. 
phus  Frederic,  59;  Gustavus  III.,  I 
71;  Gustavus  IV. ,92. 

DENMARK.— Fred.  IV.,  1699;  Cliris-  DEN.  —  Frederic  VI.,  8;  Frederic 
tian  VI.,  30;  Fred.  V.,  46;  Chris-  VII. ,30;  Christian  VIII.,  39;  Fre- 
tian  VII.,  66.  deric  VIII.,  48. 


NAPLES. — Charles  II.,   13;    Charles 
III.,  35;  Ferdinand  IV.,  59. 


NAP. — Joseph  Napoleon,  8;  Joachim 
Murat.  15;  Ferdinand  I.  (of  the 
two  Sicilies),  21;  Francis,  26;  Fer 
dinand  II. ,30. 

PORTUGAL.  — John  V.,    6;     Joseph  PORT.  — Pedro   IV.,   26;    Maria  da 
Eraanuel,50:  Maria,  77;  John VI.,      Gloria,  28. 
99. 

PRUSSIA. — Frederic  I.,  1;    Frederic  PRUS. — Frederic  W7illiam  IV.,  40. 
William  I.,  13;    Frederic  II.,  the 
Great,  40;   Frederic  Wrilliam  II., 
86;  Frederic  William  III.,  97. 


§106.] 


THE   MOON'S   AGE   AND   SOUTHING. 


335 


BIRTHS  AND  DEATHS  or  CELEBRATED  PERSONS. 


Born. 

Died.       |j 

Jioru.     !      Died. 

Solon           

B.C.  650 

it    

"    470 
"    430 
"    381 
"    356 
"    217 
"    235 
"    106 
"      70 
A.D.  274 
"     570 
"     742 
"     765 
u     

"  1160 
"   1265 
"  1304 
<  MOO 
'  1410 
'  1442 
'  1473 
'  1474 
'  1475 
'  1483 
"  1483 
•'  Mill 
"  1550 

B.C.  559 
«    479 
"    400 
"    347 
"    322 
"    323 
"    183 
"    184 
"      43 
"      10 
A.D.  337 
"     632 
"     814 
"     809 
"  1036 
"  1227 
"  1321 
"  1374 
"  1468 
"  1431 
'   1506 
«  1543 
'  1561 
'  1541 
'  1546 
"  1520 
"  1556 
"  1617 

Oliver  Cromwell  .  ... 
iMilton  

Algernon  Sidnev..  .  . 
Moliere 

A.  D.]  5!J<) 
"    1608 
"    1617 
"    1(522 
"    1623 
»    1624 
»    1632 
»    1643 
"    1646 
»    1651 
"    1672 
"    1680 
«    1681 
«    1684 
"    1694 
"    1705 
u    1707 

A.D.  1658 
"   1674 
1683 
1673 
1662 
1690 
172-3 
1727 
1716 
1715 
1725 
1760 
1760 
1753 
1778 
1790 
1783 

Confucius  

Plato  

Demosthenes  

Alexander  the  Great 

Christopher  Wren. 
Isaac  Newton  
Leibnitz..  

Scipio  Africanus..  .  . 
Cicero    

Virgil...  
Const;)  ntine  

Peter  the  Great...  . 
Handel  
Youn"      .... 

Ch;irlein;i«rne  
Haroun  Alrasehid  .. 
Canute  
Jeno-his  Khun 

Berkley 

Voltaire  
John  Wesley  
Enter 

Dante  .. 
Petrarch 

Linnaeus  
Rousseau  
Hume  

'"    1707 
«    1712 
«    1717 
•'    1723 
"    1723 
"    1723 
"    1731 
«    1737 
"    1759 
"    1769 
"    1772 

1778 
•     1778 
1776 
1780 
1790 
1792 
1800 
1794 
1794 
1821 
1S37 

Gutenberg  
Joan  of  Arc 

Columbus  

Copernicus  
Michael  Angelo  
Pizarro  
Luther   

Adam  Smith  
Joshua  Reynolds  
Cowper  

Gibbon  

Raphael  

Loyola  

Napoleon  „ 
Fourier  

Napier.  ...   

EXAMPLES. 

1.  Columbus  discovered  America  Oct.  11,  1492,  0.  S. 
What  was  the  day  of  the  week  ?  Ans.  Thursday. 

2.  On  what  day  of  the  week  was  the  commencement  of 
the  year  in  which  Henry  III.  ascended  the  English  throne  ?a 

Ans.  Friday. 

1O6.    THE  MOON'S  AGE  AND  SOUTHING. 

I.    The  Golden  Number. 

The  mean  time  of  the  moon's  revolution  around  the  earth, 
is  29.53  days.  In  19  years,  or  228  solar  months,  there  are 
235  lunar  months.  This  period  of  19  years,  is  called  a 
lunar  cycle,  and  all  the  changes  and  eclipses  of  the  moon, 
are  the  same  for  each  cycle. 

a  In  England,  until  the  year  1752,  the  year  was  considered  as  be 
ginning  on  the  25th  of  March. 


336  MISCELLANEOUS   PROBLEMS.        [ART.  XXIII. 

The  year  of  the  lunar  cycle  corresponding  to  any  given 
year,  is  called  the  GOLDEN  NUMBER  for  that  year.  It  is 
found  by  the  following  rule  : 

Add  1  to.  the  given  year,  and  divide  l>y  19.  The  re 
mainder  will  be  the  golden  number.  (If  the  remainder  is 
0,  the  golden  number  is  19.) 

Required  the  golden  number  for  A.  D.  1853.  if  Jl  = 
97fJ,  Am.  11. 

II.    The  Epact. 

The  Epact  is  the  moon's  age  at  the  beginning  of  the  year. 
It  generally  either  increases  by  11,  or  diminishes  by  19, 
every  year,  and  never  exceeds  29.  It  is  found  by  the  fol 
lowing  rule : 

Divide  the  given  year  by  19,  and  multiply  the  remainder 
by  11.  The  product  will  be  the  epact,  if  it  does  not  exceed  29. 

If  the  product  exceeds  29,  divide  it  by  30,  and  the  re 
mainder  will  be  the  epact. 

The  epact  for  1845 :  if  Jl  =  97-^;  2  X  11  =  22.  Ans.  22. 
The  epact  for  1857 :  iff  =  97 J| ,',  14  X  11  =  154;  */J 
—  534<j.     Ans.  4. 
The  epact  for  1862  :  -1  f  j*  =  98.     Ans.  0. 

III.    The  Number  or  Epact  for  the  Month. 

The  Number  for  any  month,  shows  what  the  moon's  age 
would  be  at  the  beginning  of  that  month,  provided  it  was 
new  moon  on  the  first  of  January.  It  is  found  by  the  fol 
lowing  rule : 

Divide  the  number  of  days  in  the  preceding  months, 
(reckoning  from  the  beginning  of  the  year ,)  by  29.53,  and, 
the  nearest  whole  number  to  the  remainder,  is  the  epact  or 
number  for  the  month,  required. 

O1    __[__  OQ      i       O1 

The  epact  for  April.     In  common  years,  -     ^  53 = 


§106.]  THE   MOON'S   AGE   AND    SOUTHING.  837 

rni  f  -i         T        1  31+29-|-3i 

Ihe   epact   for   April,     in   leap  years,  —  ~9o~   ---  — 


IY.   The  Moon's  Age. 

The  Moon's  Age,  is  the  number  of  days  that  have  elapsed 
since  new  moon.  It  never  exceeds  30.  It  is  found  by  the 
following  rule  : 

To  the  epact  for  the  year,  add  the  number  for  the  month, 
and  the  day  of  the  month,  and  divide  by  30.  The  remain 
der  will  be  the  moon's  age. 

What  was  the  moon's  age;  Aug.  18th,  1849  ?     6"h^18 

29 
=»gQ-         Ans.  29  days. 

Y.    The  Moon's  Southing. 

The  SOUTHING  is  the  time  when  the  moon  is  on  the 
meridian.  It  is  found  by  the  following  rule: 

Multiply  the  moon's  age  by  .8,  and  the  product  will  be 
the  hours  past  noon.  If  the  hours  exceed  12,  subtract  12, 
and  the  remainder  will  represent  the  time  after  midnight. 

Find  the  time  of  the  moon's  southing,  for  July  4th,  1776. 

Epact  9  ;  number  for  the  month  5  ;  moon's  age  18.  18 
X  .8  =  14.  4h.  =  2h.  24m.  after  midnight,  moon  south. 

EXAMPLES. 

1.  Find  the  Golden  Number  for  the  years  of  accession1  of 
each  of  the  monarchs  of  Great  Britain.       Ans.  Jas.  I.  8  ; 

Chas.  I.  11  ;  Cromwell,  1  ;  Chas.  II.  8  ;  &c. 

2.  Find  the  epact  for  the  years  of  accession  of  William 
the  Conqueror,  and  of  each  of  the  ten  following  monarchs  of 
England.  Ans.  Wm.  I.  22;  Wm.II.  14;  Hen.  I.  7; 

Stephen  15,  &c. 


a  See  Table  of  Contemporary  Princes. 

22 


338  MISCELLANEOUS   PROBLEMS.       [ART.  XXIIT. 

3.  Find  the  moon's  number  for  each  month  of  the  year, 
both  for  common  years,  and  for  leap  years. 

Ans.  common  years;  Jan.  0;  Feb.  1;  Mar.  29;  &c. 
leap  years;  "     0;     "      1;      "       l;&c. 

4.  What  was  the  moon's  age  on  the  day  of  the  battle  of 
Bunker  Hill,  June  17th,  1775  ? 

5.  The  French  army  commenced  its  retreat  from  Moscow, 
Oct.  18th,  1812.     At  what  time  on  that  day  was  the  moon 
on  the  meridian  1 

1O7.    MENSURATION. 

In  the  following  formulas,  a  represents  the  area ;  al  the 
altitude ;  b  the  base ;  br  the  breadth ;  c  the  circumference ; 
ca  the  conjugate  axis,  (of  an  ellipse) ;  cs  the  convex  sur 
face  ;  d  the  diameter ;  fa  the  fixed  axis,  (of  a  solid  of  revo 
lution)  ;  h  the  hypothenuse ;  he  the  height ;  I  the  length ; 
Ib  the  lower  base,  (of  a  frustum);  p  the  perpendicular; 
ra  the  revolving  axis ;  s  the  solidity ;  sh  the  slant  height, 
(of  a  cone,  frustum,  &c.) ;  ub  the  upper  base. 

1.  THE  PARALLELOGRAM,     a  —  lxal. 

2.  THE  TRIANGLE,     a  =  ^_a£ 

To  find  the  area  of  a  triangle,  when  the  three  sides  are 
given : 

I.  Add  the  three  sides  together,  and  take  half  their  sum. 

II.  From  this  half  sum  take  each  side  separately. 

III.  Form  the  continued  product  of  the  half  sum  and  the 
three  remainders,  and  extract  the  square  root  of  that  product. 


3.  THE  RIGHT  ANGLED  TRIANGLE.     h=  ^b2  +  p2. 

4.  THE  CIRCLE,    c  =  3.1416  X  d;  a  =  .7854  x  d2;  a  = 
.07958  x  c2. 

5.  THE  ELLIPSE,     a  =  ta  X  ca  X  .7854. 


§  107.]  MENSURATION.  339 

6.  THE  PARALLELOPIPEDON.     s  =  I  X  br  x  he. 

7.  THE  PRISM,     s  =  al  X  a  of  b. 

8.  THE  CYLINDER,     cs  =  al  X  c  of  b;  s  =  al  X  a  of  b. 

9.  THE  CONE  AND  PYRAMID,    cs  =  sh  X  c  of  5;  s  = 
al  X  a  of  b 


°f 


+ 


s  = 


10.  THE  FRUSTUM.     cs  = 

a  of  M&  +  a  of  Ib  +  V«  of  ub  X  a  of  /6  )  X  al 


11.  THE  SPHERE,     cs  =  c  x  c?;   s  =  .5236  X  d3;  s  = 
.01689  x  c3. 

12.  THE  ELLIPSOID,    s  =/a  x  m2  X  .5236. 

13.  THE  POLYGON.     To  find  the  area  of  a  regular  poly 
gon  when  one  of  its  sides  is  given :  Multiply  the  square  of 
the  side,  by  the  multiplier  standing  opposite  the  name  of 
the  polygon  in  the  following  table.     The  radius  of  the  in 
scribed  circle,  is  found  by  multiplying  the  side  by  the  proper 
number  in  the  right  hand  column. 


No.  of 
Sides. 

Names. 

Multipliers. 

Radius  of  In 
scribed  Circle. 

3 

Triangle 

.433013 

.288675 

4 

Square 

1.000000 

.500000 

5 

Pentagon 

1.720477 

.688191 

6 

Hexagon 

2.598076 

.866025 

7 

Heptagon 

3.633912 

1.038262 

8 

Octagon 

4.828427 

1.207107 

9 

Nonagon 

6.181824 

1.373739 

10 

Decagon 

7.694209 

1.538842 

11 

Hendecagon 

9.365640 

1.702844 

12 

Dodecagon 

11.196152 

1.866025 

14.  THE  POLYEDRON.  To  find  the  surface  of  a  regular 
polycdron :  Multiply  the  square  of  the  linear  edge,  by  the 
tabular  number  in  the  column  of  surfaces. 


340  MISCELLANEOUS   PROBLEMS.       [ART.  XXIII. 

To  find  the  solidity  of  a  regular  polyedron :  Multiply  the 
cube  of  the  linear  edge,  by  the  tabular  number  in  the  column 
of  solidities. 


No.  of 
Sides. 

Names. 

Surfaces. 

Solidities. 

4 

6 
8 
12 
20 

Tetraedron 
Hexaedron 
Octaedron 
Dodecaedron 
Icosaedron 

1.73205 
6.00000 
3.46410 
20.64578 
8.66025 

.11785 
1.00000 
.47140 
7.66312 
2.18169 

EXAMPLES. 

1.  By  the  second  method  of  Analysis,  (§§  57  and  59,)  de 
termine  the  formulas  for  finding  I,  and  al,  in  a  parallelogram. 

Ans.  b  =  a  -f-  al;  al  =  a  ~-  b. 

2.  What  are  the  formulas  for  I  and  al,  in  a  triangle? 

3.  Required  the  formulas  for  b  and  p,  in  a  right-angled 
triangle.  Ans.  I  =  VA2-/;  P  =  V^2  -  b2. 

4.  Find  two  formulas  for  d,  and  an  additional  formula 
for  c,  in  a  circle.  Ans.  d  =  c  -j-  3.1416;  d  = 


;  c=  Ja-r-.  07958. 

5.  Obtain  formulas  for  ta  and  ca  in  an  ellipse. 

6.  What  are  the  formulas  for  I,  br}  and  he  in  a  parallelo- 
pipedon  ? 

7.  Find  formulas  for  al  and  a  of  b  in  a  prism. 

8.  Obtain  formulas  for  c  of  b  and  a  of  b}  and  two  formulas 
for  al  in  a  cylinder. 

9.  Obtain  formulas  for  sh,  al,  c  of  b,  and  a  of  b  in  a  cone 
or  pyramid. 

10.  Find  two  formulas  for  c  and  for  d  in  a  sphere. 

Partial  Ans.  c=tys  +  .01689. 


§  107.]  MENSURATION.  341 

11.  Required  formulas  for  c  of  w5,  c  of  Ibj  sTi,  and  a?,  in 
the  frustum  of  a  pyramid,  or  of  a  cone. 

Ans.  c  of  ub  =  ?ff  —  c  of  Ib;  c  of  Ib  =  ?ls-c  of  ub : 
sh  sh 

sh  =  Ics  -f-  (c  of  ub  +  c  of  Ib) ;  al  =  3s  -^  (a  of 
•w6  -f-  a  of  Z6  +  \/tt  of  ^  X  a  of  £6). 

12.  What  are  the  formulas  for  fa  and  ra  in  an  ellipsoid? 
^bis.  fa  =  s  -f-  (.5236  X  ra2);  ra  =  Vs-r-(.5236  X  fa.) 

13.  The  solidity  of  a  prism  is  516  c.  ft.,  and  the  alti 
tude  27  Jft.     What  is  the  area  of  its  base  ? 

Ans.  18-Jf  sq.  ft. 

14.  The  convex  surface  of  a  pyramid  contains  47  sq.  ft., 
and  the  circumference  of  its  base  is  3£ft.     Required  its 
slant  height?  Ans.  28jft. 

15.  The  solidity  of  a  frustum  is  1447i  c,  ft.;  the  area  of 
the  upper  base  49  sq.  ft. ;  the  area  of  the  lower  base  81  sq. 
ft.     What  is  its  altitude?  Ans.  22£ft. 

16.  The  solidity  of  a  cylinder  is  28  c.  ft.,  and  the  altitude 
14ft.     What  is  the  diameter  of  the  base  ?     Ans.  1.59ft. 

17.  Required  the  radius  of  a  sphere  whose  solidity  is 
75  c.  ft.  Ans.  2.616ft. 


18.  The  fixed  axis  of  an  ellipsoid  measures  10ft.;  and 
its  solidity  is  80  c.  ft.     What  is  the  length  of  its  revolving 
axis?  Ans.  3.9088ft. 

19.  The  area  of  a  regular  dodecagon  is  47ft.     Required 
the  length  of  each  side,  and  the  radius  of  the  inscribed 
circle.  Ans.  2.0488ft.;  3.823ft. 

20.  What  is  the  area  of  a  triangle,  whose  sides  measure 
respectively  16  rods,  27  rods,  and  19  rods  ? 

Ans.  3R.  29.398r. 


342  MISCELLANEOUS   PROBLEMS.        [.AJIT.  XXIII. 

21.  A  regular  icosaedron  contains  81  c.  ft.  What  is  the 
length  of  each  of  its  linear  edges?*  Ans.  3.336ft. 

1O8.    TONNAGE  OF  VESSELS. 

The  accurate  determination  of  the  tonnage  of  any  vessel, 
is  a  very  difficult  problem.  The  best  method  probably,  is, 
to  divide  the  vessel  into  a  number  of  sections,  and  to  de 
termine  by  numerous  measurements,  the  average  length, 
breadth,  and  depth  of  each  section.  The  solid  contents  can 
thus  be  found  in  cubic  feet,  and  by  dividing  the  number  of 
cubic  feet  by  40,b  the  answer  will  be  obtained  in  tons.  The 
accuracy  of  the  result,  will  depend  on  the  number  of  meas 
urements  that  are  taken. 

This  process  is,  however,  a  very  tedious  one,  and  numerous 
rules  have  been  framed,  in  order  to  abridge  the  labor  of 
computation.  Of  these  rules,  the  following  are  the  most 
frequently  employed. 

I.    Carpenters'  Rule. 

Measure  the  length,  breadth,  and  depth,  all  in  feet.  Divide 
the  continued  product  of  the  three  dimensions  by  95,  and 
the  quotient  will  be  the  tonnage. 

If  the  vessel  is  double-decked,  half  the  breadth  is  taken 
as  the  depth. 

II.    United  States  Government  Rule. 

If  the  vessel  is  single-decked,  take  the  length  from  the 

fore  part  of  the  main  stem  to  the  after  part  of  the  stern  post 

above  the  upper  deck,  the  breadth  at  the  broadest  part  above 

the  main  wales,  and  the  depth  from  the  under  side  of  the 

a  Questions  similar  to  the  foregoing,  may  be  multiplied  by  the 
teacher,  to  any  extent  that  may  seem  desirable. 

b  Dividing  by  40  will  give  the  actual  tonnage.  Each  of  the  following 
rules  gives  a  result  considerably  less  than  the  true  tonnage.  In  freight 
ing  by  the  ton,  the  owner  may  charge  either  by  the  ton  of  weight,  or 
the  ton  of  measure,  whichever  will  yield  the  largest  tonnage. 


§108.]  TONNAGE   OF   VESSELS.  343 

deck  to  the  ceiling,  in  the  hold.  From  the  length  deduct 
I  of  the  breadth,  multiply  the  remainder  by  the  breadth, 
and  that  product  by  the  depth.  Divide  the  last  product  by 
95,  and  the  quotient  will  be  the  government  tonnage. 

If  the  vessel  is  double-decked,  substitute  half  of  the 
breadth  for  the  depth,  and  proceed  as  directed  for  single- 
decked  vessels. 

III.  British  Government  Rule. 

Divide  the  length  of  the  upper  deck  between  the  after  part  of 
the  stem  and  the  fore  part  of  the  stern-post,  into  6  equal  parts. 
At  the  foremost,  the  middle,  and  the  aftermost  of  these  points  of 
division,  measure  in  feet,  and  decimals  of  a  foot,  the  depths  from 
the  under  side  of  the  upper  deck  to  the  ceiling  at  the  limber 
strake.  (In  the  case  of  a  break  in  the  upper  deck,  the  depths  are  to 
be  measured  from  aline  stretched  in  a  continuation  of  the  deck.) 
Divide  each  of  the  three  depths  into  5  equal  parts,  and  measure 
the  inside  breadths  at  the  following  points,  viz  :  at  \  and  5  below 
the  upper  deck  of  each  of  the  extreme  depths,  and  at  f  and  | 
below  the  upper  deck  of  the  midship  depth.  At  half  the  midship 
depth  measure  the  length  from  the  after  part  of  the  stem  to  the 
fore  part  of  the  stern-post. 

To  twice  the  midship  depth  add  the  two  extreme  depths,  for  the 
sum  of  the  depths.  Add  the  upper  and  lower  breadths  at  the  fore 
most  division,  3  times  the  upper  breadth  and  the  lower  breadth 
at  the  midship  division,  and  the  upper  and  twice  the  lower  breadth 
at  the  after  division,  for  the  sum  of  the  breadths.  Multiply  the 
sum  of  the  depths  by  the  sum  of  the  breadths,  and  the  product 
by  the  length,  and  divide  the  final  product  by  3500,  which  will 
give  the  number  of  tons  for  register.  If  the  vessel  has  a  half- 
deck,  or  a  break  in  the  upper  deck,  measure  the  inside  mean 
length,  breadth,  and  height  of  such  part  thereof  as  may  be  in 
cluded  within  the  bulk-head.  Divide  the  continued  product  of 
these  three  measurements  by  92.4,  and  the  quotient  will  be  the 
number  of  tons  to  be  added  to  the  result  as  above  found.a 

a  Previous  to  Jan.  1st,  1846,  the  British  rules  for  estimating  tonnage 
were  similar  to  our  own.  But  this  rule  led  to  the  building  of  ships  of 
forms  improper  for  the  purpose  of  safe  navigation,  in  order  that,  by 
measuring  less  than  their  real  burden,  they  might  evade  a  part  of  the 


344  MISCELLANEOUS   PROBLEMS.      [ART.  XXHI. 

If  the  vessel  is  laden,  measure  the  length  on  the  upper  deck  be 
tween  the  after  part  of  the  stem  and  the  fore  part  of  the  stern- 
post,  the  inside  breadth  on  the  under  side  of  the  upper  deck  at 
the  middle  point  of  the  length,  and  the  depth  from  the  under  side 
of  the  upper  deck  down  the  pump-well  to  the  skin.  Divide  the 
continued  product  of  these  three  dimensions  by  130,  and  the 
quotient  will  be  the  register  tonnage. 

In  open  vessels,  the  depths  are  to  be  measured  from  the  upper 
edge  of  the  upper  strake. 

EXAMPLES. 

1.  Find  by  rule  1,  the  tonnage  of  a  double-decked  vessel 
163  feet  long,  and  31  feet  wide.  Ans.  824?  tons. 

2.  Find  by  rule  2,  the  government  tonnage  of  a  double- 
decked  vessel  180  ft.  long,  and  32ft.  wide. 

Ans.  866.6  tons. 

3.  Required  the  British  government  tonnage  of  a  vessel 
with  the  following  measurements  :  length  at  half  the  mid 
ship  depth  175ft. ;  depths,  at  the  foremost  point  of  division 
29ft.,  at   the  middle  point  30ft.,  at   the  aftermost   point 
28ft. ;  breadths,  at  the  J  and  |  depths  of  the  foremost  di 
vision  33  and  24ft.,  at  the  f  and  -J  depths  of  the  midship 
division  39  and  36ft.,  at  the  J-  and  -J  depths  of  the  after 
most  division,  35  and  29ft. ;  half-deck,  length  75ft.,  mean 
breadth  35ft.,  height  7ft.  Ans.  1971.41  tons. 

1OO.     G-AUGING. 

The  easiest  way  of  finding  the  contents  of  casks,  is  by 
the  diagonal  rod.  The  contents  given  by  the  rod  are  only 
correct  for  casks  of  the  most  common  forms. 

The  same  result  as  the  rod  would  give,  may  be  found  in 
the  following  manner : 

duties.  The  method  now  employed,  gives  the  tonnage  of  all  ships, 
however  built,  with  tolerable  accuracy,  and  therefore  removes  the 
temptation  to  build  vessels  of  an  unsuitable  form. 


§109.]  GAUGING*.  845 

Take  any  rod,  and  putting  it  in  at  the  bung,  extend  it  to 
the  opposite  corner  of  either  head,  and  measure  the  distance 
in  inches.  Extend  it  in  a  similar  manner  to  the  other  head, 
and  measure  the  distance.  Half  the  sum  of  these  two  dis 
tances,  will  be  the  diagonal.  Then, 

Multiply  the  cube  of  the  diagonal  by  <j]-JJ<>  =  ?<ftny. 
The  quotient  will  be  the  contents  in  imperial  gallons.  For 
wine  gallons,  substitute  173;  for  beer  gallons,  141.6;  for 
dry  gallons,  148.5,  in  the  place  of  144. 

In  order  to  determine  the  contents  with  accuracy,  casks 
are  usually  divided  into  four  varieties,  according  to  the 
degree  of  their  curvature.  In  each  of  the  following  formu 
las  for  those  four  varieties,  D  denotes  the  bung  diameter, 
d  the  head  diameter,  and  a  the  length  of  the  cask. 

I.  When  the  cask  is  formed  like  the  middle  zone  of  a 

spheroid. 

.0009442  X  a  X  (2  D2  +  d?)  =  contents  in  imperial  gal 
lons.  For  the  contents  in  wine  gallons,  substitute  .0011333 ; 
for  beer  gallons,  .0009284;  for  dry  gallons,  .000974,  in 
the  place  of  .0009442. 

II.  When  the  cask  is  formed  like   the  middle  zone  of  a 

parabolic  spindle. 

.0001888  x  a  X  (8  D2  +  4  D  X  d  +  Bd2)  =  contents  in 
imperial  gallons.  For  wine,  beer,  or  dry  measure,  sub 
stitute  .0002267,  .0001856,  .000195,  in  the  place  of 

.0001888. 

III.  When  the  cask  is  formed  like  two  equal  frustums  of  a 

parabolic  conoid. 

.0014163  X  ax  (D2  +  d2)  =  contents  in  imperial  gallons. 
For  wine,  beer,  or  dry  measure,  substitute  .0017,  .0013926, 
.001461,  in  the  place  of  .0014163, 


346  MISCELLANEOUS    PROBLEMS.       [ART.  XXIII. 

IV.    When  the  cask  is  formed  like  two  equal  frustums  of 

a  cone. 

.0009442  X  a  X  (D2  +  D  X  d  +  d1}  =  contents  in  impe 
rial  gallons.  For  the  other  measures,  substitute  as  in 
Case  I. 

The  following  method  is  more  accurate  than  either  of  the 
foregoing,  when  the  diameter  midway  between  the  head 
and  the  bung,  can  be  accurately  determined : — 

Add  the  squares  of  the  head  diameter,  of  the  bung 
diameter,  and  of  twice  the  middle  diameter.  The  sum, 
multiplied  by  .0004721  times  the  length,  gives  the  contents 
in  imperial  gallons.  For  wine  gallons,  substitute  .0005667; 
for  beer  gallons,  .0004642;  for  dry  gallons,  .000487,  in 
the  place  of  .0004721. 

If  the  staves  are  of  uniform  thickness,  the  middle  diameter 
may  be  found  by  measuring  the  circumference,  dividing  by 
3.1416,  and  subtracting  twice  the  thickness  of  the  staves  from 
the  result. 

EXAMPLES. 

1.  The  diagonal  of  a  cask  is  32  inches.     Required  its 
contents  in  wine  gallons. 

Ans.  (32)3  X  173  -~  64000  =  88.576  gallons. 

2.  Find  the  contents  in  imperial  gallons,  of  casks  of  each 
of  the  four  varieties,  the  length  of  each  cask  being  40 
inches,  and  the  diameters  30  and  36  inches. 

Ans.  1st  var.  .0009442  X  40  X  (2  X  362  +  302)  = 

131.886  gallons. 
2d  var.  .0001888  X  40  x  (8x362  +  4x36x 

30  +  3  X  302)  ==  131.314  gallons. 
3d  var.  .0014163  X  40  X  (362  +  302)=124.408 

gallons. 

4th  var.  .0009442  x  40  X  (362 +36x30  +  302) 
=  123.628  gallons. 


§110.]  MISCELLANEOUS   EXAMPLES.  347 

3.  The  head  diameter  of  a  cask  is  25  in.,  the  bung  dia 
meter  30  in.,  the  middle  diameter  28  in.,  and  the  length 
36  in.     Required  the  contents  in  beer  measure. 

Ans.  (252  +  302  +  562)  X  .0004721  X  36  =  79.216 
gallons. 

4.  The  circumference  of  a  cask  at  the  bung  is  113  in., 
at  the  head  91  in.,  and  at  a  point  midway  between  the  head 
and  the  bung  106  in.     Required  the  contents  in  each  meas 
ure,   the   length   being  40in.,  and   the   thickness  of  the 
staves  £  in.a  Ans.  119.16  imperial  gallons. 

143.03  wine  gallons. 
117.16  beer  gallons. 
122.92  dry  gallons. 

1 1 0.    MISCELLANEOUS  EXAMPLES. 

1.  If  the  multiplicand  is  7,  and  the  product  2,  wha<   is 
the  multiplier  ? 

2.  The  dividend  is  1,  and  the  quotient  8 ;  what  is  the 
divisor  ? 

3.  The  dividend  is  6|,  and  the  quotient  18§;  what  is 
the  divisor? 

4.  The  sum  of  two  numbers  is  71,  and  one  of  the  num- 

o  / 

bers  is  4.759 ;  what  is  the  other  ? 

5.  The  difference  of  two  numbers  is  13|,  and  the  greater 
number  is  29.43 ;  what  is  the  less  ? 

6.  The  sum  of  two  numbers  is  7-fg,  and  their  difference 
is  6g3T ;  what  are  the  numbers  ? 

a  When  the  circumferences  are  given,  it  is  not  necessary  to  find 
each  diameter  separately.  We  may  proceed  with  the  circumferences 
precisely  as  with  the  diameters,  until  we  obtain  the  sum  of  the 
squares.  This  sum  should  be  divided  by  the  square  of  3.1416 
(9.8696),  and  the  quotient  will  be  the  sum  of  the  squares  of  the 
diameters.  A  deduction  should  be  made  from  each  outside  circum 
ference,  of  six  times  the  thickness  of  the  staves,  and  the  remainder 
will  be  nearly  equivalent  to  the  inside  circumference. 


348  MISCELLANEOUS   PROBLEMS.       [ART.  XXIII 

1*7 
Li 


7.  What  number  must  be  subtracted  from  39  1  to  leave 


8.  What  number  must  be  added  to  23y\  to  make  47.432  ? 

9.  What  number  must  be  multiplied  by  3|,  and  the  pro 
duct  divided  by  7.365  to  give  8|  as  a  quotient  ? 

10.  What  is  the  difference  between  -f^  and  $f  of  7T. 
15cwt.  3qr.  ? 

11.  What  is  the  difference  between  28  miles,  and  27m. 
7fur.  39r.  5yd.  2ft.  11.9in.? 

12.  Reduce  4  of  9rn.  7fur.  39r.  5yd.  2ft.  to  inches. 

13.  If  from  a  purse  containing  £35  7s.  lid.,  I  pay  to 
each  of  15  laborers,  £1  9s.  8|d.,  how  much  will  be  left? 

14.  Find  the  sum,  the  difference,  and  the  product  of 
874.91  and  42TY 

15.  In  128|lb.  of  water,  there  is  14|lb.  of  hydrogen. 
How  much  hydrogen  is  there  in  273|  Ib.  of  water  ? 

16.  A  grocer  sold  17cwt.  3qr.  17  Ib.  of  sugar,  at  6|  cents 
a  pound,  receiving  in  exchange  20  barrels  of  flour,  at  §4| 
per  barrel,  and  the  balance  in  money.     How  much  money 
did  he  receive  ? 

17.  From  |  of  3T.  17cwt.,  subtract  T4g  of  7T.  3cwt.  Iqr. 
18  Ib. 

18.  What  will  be  the  freight  of  17f  cwt.  for  89f  miles, 
if  $7.63  be  paid  for  carrying  HT3g  tons  9|  miles? 

19.  How  many  raisins,  at  8|  cents  a  pound,  must  be 
given  in  exchange  for  163gal.  2qt.  3gi.  of  molasses,  at  $.37| 
per  gallon  ? 

20.  Bought  16cwt.  3qr.  16  Ib.  of  rice,  at  84.00  percwt., 
and  9cwt.  2qr.  51b.  of  pearl  barley,  at  $4.37|  per  cwt.  How 
much  would  be  gained  on  the  whole,  by  selling  each  at  4} 
cents  a  pound  ? 


§110.]  MISCELLANEOUS   EXAMPLES.  349 

21.  If  63fyd.  of  broadcloth  cost  $255,  at  what  price 
must  it  be  sold  per  yd.,  in  order  to  gain  $25.50  ? 

22.  A  hogshead  of  sugar  at  $7.00  per  cwt.  cost  $43.75. 
What  did  it  weigh  ? 

23.  How  much  shalloon  that  is  f  yd.  wide,  will  line  14| 
yards  of  cloth,  that  is  l|yd.  wide  ? 

24.  What  is  the  price  of  16  boxes  of  raisins,  each  hold 
ing  9§lb.,  at  9-J-  cents  per  lb.? 

25.  How  much  money,  that  is  9  per  cent,  below  par,  will 
pay  a  debt  of  $187.50  ? 

26.  If  9  men  mow  10  acres  of  grass  in  a  day,  how  much 
will  11  men  mow  in  2J  days? 

27.  In  what  time  will  $150  gain  $6.37|,  at  6  per  cent., 
simple  interest  ? 

28.  When  molasses  is  31J  cents  a  gallon,  how  many 
hogsheads,    each    holding    97gal.    3qt.,    can    I    buy  for 
$366.56^  ? 

29.  What  will  be  the  price  of  7  bales  of  sheeting,  each 
bale  containing  9  pieces,  and  each  piece  measuring  30|yd., 
if  26yd.  cost  $2.92 '  ? 

30.  Bought  39i  bushels  of  potatoes  for  $12.87^-.     At 
what  price  per  bushel  must  they  be  sold,  in  order  to  gain 
15  per  cent.  ? 

31.  What  is  the  interest  of  $9431  for  3yr.  7mo.  13dy., 
at  7  per  cent.  ? 

32.  How  much  may  a  man  spend  per  day,  whose  income 
is  $500  a  year,  after  deducting  T5<j  per  cent,  for  taxes  ? 

33.  The  words  in  Johnson's  Dictionary  have  been  classi 
fied  as  follows:*  Articles,  3;   nouns,  20409;   adjectives, 

* 'Gregory. 


350  MISCELLANEOUS   PROBLEMS.       [ART.  XXIII. 

9053 ;  pronouns,  41 ;  verbs  active  5445,  neuter  2425,  pas 
sive  1,  defective  5,  auxiliary  1,  impersonal  3 ;  verbal  noun, 
1 ;  participles,  38 ;  participial  adjectives,  125 ;  participial 
nouns,  3;  adverbs  in  ly,  2096;  other  adverbs,  496;  pre 
positions,  69;  conjunctions,  19;  interjections,  68.  What 
per-centage  of  the  entire  number  belongs  to  each  part  of 
speech  ? 

34.  Owing  to  the  curvature  of  the  earth,  the  difference 
of  level  in  1  mile  is  8  inches.     What  would  be  the  differ 
ence  in  J  mile,  the  level  varying  as  the  square  of  the  dis 
tance  ?    in  5 1  miles  ?  Ans.  2in. ;  222Lft. 

35.  Suppose  a  school  of  180  boys  to  breathe  20  times 
each  per  minute,  requiring  for  each  respiration  30  c.  in.  of 
air,  what  amount  of  carbonic  acid  would  they  produce  in 
two  sessions,  of  3  hours  each,  estimating  that  5  per  cent, 
of  the  air  inhaled  is  changed  into  an  equal  volume  of  car 
bonic  acid  ?  Ans.  1125  c.  ft. 

36.  A  vessel  of  400  tons  has  a  keel  48  feet  long.  What 
length  of  keel  has  a  vessel  of  750  tons,  that  is  built  on  the 
same  model?  Ans.  59.19ft. 

37.  There  are  two  cannon  balls,  one  weighing  28  pounds, 
and  the   other  9   pounds.     What  is  the  diameter  of  the 
greater,  that  of  the  less  being  5  inches  ? 

Ans.  7.3in.,  nearly. 

38.  In  an  establishment  in  Lowell  50  cows  were  kept,  of 
which  the  average  number  giving  milk  was  35.     Each  cow 
consumed  annually  4.18  tons  of  hay  at  318.50  per  ton,  and 
green  vegetables  worth  $20.36.     What  was  the  loss  in  two 
years,  exclusive  of  the  amount  expended  for  attendance,  the 
whole  amount  of  milk  obtained  being  99705  quarts,  worth 
five  cents  a  quart  ? 

39.  Bought  6  bales  of  cinnamon,  weighing  gross  4cwt 
3qr.  2  lb.,  tare  9  lb.?  at  811  per  cwt.?  and  paid  charges  for 


f  • 

§110.]  MISCELLANEOUS   EXAMPLES.  351 

freight,  duties,  &c.  $11.43f .     What  rate  per  cent,  would 
be  gained  or  lost,  by  selling  the  whole  at  $.15  per  Ib.  ? 

Ans.  25  per  cent,  gained. 

40.  Find  the  value  of  4  cases  of  gum  tragacanth,  at  £20 
8s.  per  cwt.,  duty  25  per  cent,  ad  valorem,  the  cases  weigh 
ing  as  follows,  viz : 

cwt.  qr.    Ib. 

No.  15  gross  217     tare  41  Ib.  ^ 
No.  16     «     2     1     11       «    42  « 

No.  17     "21       7       "     40"    r  Draft  2  Ib.  per  case. 
No.  18     "     2     0     27      "    39  "    J 

Ans.  £196    5s.    2  id. 

41.  Required  the  amount  of  commission  at  2J  per  cent., 
and  brokerage  at  J  per  cent.,  and  the  net  proceeds  of  50 
bags  of  cotton,  weighing  gross  115cwt.    Iqr.  11  Ib.,  draft 
1  Ib.  per  bag,  tare  4  Ib.  per  cwt.,  the  whole  being  sold  at 
7  Jets,  per  Ib.     Ans.  Commission  and  brokerage  $27. 92. 

Net  proceeds  $902.68. 

42.  In  the  British  Sanitary  Report  of  1843,  it  is  stated 
that  the  proportionate  numbers  of  the  population  at  different 
ages,  in  the  United  States,  and  in  England  and  Wales,  were 
as  follows  : — 

United  States.    England  &  Wales. 

Under  5  years  1744  1324 

5  and  under  10  1417  1197 

10    «         "     15  1210  1089 

15    "        "     20  1091  997 

20   "        "     30  1816  1780 

30    "        «'<     40  1160  1289 

40   "        "     50  732  959 

50   "        "     60  436  645 

60   "        "     70  245  440 

70    "        "     80  113  216 

80   "        "     90  32  59 

90  and  upwards  4  5 
Required   the   average   age,  both    in    England   and   in 


352  MISCELLANEOUS   PROBLEMS.        [ART.  XXIII. 

America,  of  all  the  inhabitants; — of  all  above  15; — above 
20 ;— above  50. 

43.  The  commissioners  of  a  certain  county  are  about 
building  a  new  court  house,  which  will  cost  $75000.     They 
hire  money  for  the  purpose  at  an  annual  interest  of  5  per 
cent.,  and  propose  to  pay  the  debt  thus  incurred  by  50  equal 
annual  instalments.     What  amount  must  be  paid  each  year? 

Ans.  $4108.25. 

44.  Three  men  bought  a  grindstone  50  inches  in  diameter, 
for  which  A.  paid  75  cents,  B.  $1.50,  and  C.  $2.00.     What 
part  of  the  diameter  ought  each  to  wear  away,  allowing  the 
diameter  of  the  axle  to  be  2  inches  ? 

Ans.  A.  4.62in. ;  B.  11.05in. ;  C.  32.33in.a 

45.  An  estate  of  $20000  is  to  be  divided  between  two 
sons  in  the  following  manner :  the  elder  is  to  receive  $100 
the  first  month,  $300  the  second  month,  &c.,  in  arithmetical 
progression,  and  the  younger  is  to  receive  $1000  per  month, 
until  the  whole  is  paid.    What  is  the  share  of  each,  and  how 
long  will  they  be  in  receiving  it  ?    Ans.  $10000 ;  lOmo. 

46.  A  ladder  standing  upright  against  a  wall  reaches  the 
top,  but  the  foot  being  removed  12  feet  from  the  wall,  it 
reaches  to  a  point  6  feet  from  the  top.     Required  the  length 
of  the  ladder,  and  the  height  of  the  wall.        Ans.   15ft. 

47.  A  block  of  stone  12ft.  long,  3ft.  wide,  and  2ft.  thick, 
is  to  be  floated  on  a  pine  raft,  which  is  20ft.  long,  and  6ft. 
wide.     What  must  be  the  depth  of  the  raft,  in  order  that 
it  may  float  4  inches  above  the  water,  the  specific  gravity 
of  the  pine  being  575,  and  the  sp.  grav.  of  the  stone  2500  ? 

Ans.  4ft.  3. 76  -f  in. 

48.  Wishing  to  estimate  the  height  of  a  hill  which  is  5 
miles  distant,  I  hold  a  foot  rule  at  the  distance  of  2ft.  from 
my  eye,  and  find  that  1  inch  on  the  rule  intercepts  the  rays 

*  This  answer  is  obtained  by  supposing  that  A.  uses  the  stone  first, 
B.  second,  and  C.  last.     The  question  admits  of  five  other  solutions. 


§110.]  MISCELLANEOUS    EXAMPLES.  353 

from  the  top  of  the  hill  and  from  the  horizon.     What  is 
the  height  of  the  hill  ?  Ans.  1100ft. 

49.  What  is  the  distance  of  a  thunder  cloud,  if  three 
seconds  elapse  between  the  flash  and  report  ? 

Ans.  3270ft. 

50.  If  the  travel  over  a  hill,  by  friction  and  gravity  causes 
8000  days'  work  of  a  horse,  at  75  cents  per  day,  which  can  be 
partially  avoided  by  a  road  along  the  base  of  the  hill,  the 
travel  over  which  would  require  only  4000  days'  work,  and  if 
the  new  road  will  require  an  extra  annual  outlay  of  $600  for 
repairs,  how  much  can  be  saved  by  expending  812000  in 
making  the  improvement,  computing  interest  at  5°]0  ? 

Ans.  $36000. 

51.  If  an  engine  has  sufficient  force  to  draw  100  tons  over 
level  ground,  what  additional  power  must  be  exerted  on  an 
ascending  grade  of  40ft.  per  mile  ? 

Ans.  15cwt.  163f  j^ 

52.  What  is  the  power  of  a  steam  engine  with  a  cylinder 
40  inches  in  diameter,  making  the  usual  estimate  of  the 
effective  force  of  the  steam  and  the  stroke  of  the  piston  ? 

Ans.  64  horse  power. 

53.  What  is  the  amount  of  pressure  on  a  dam  150ft.  by 
18ft.,  the  average  depth  of  water  being  6ft.  ? 

Ans.  1012500  Ib. 

54.  My  expenditure  having  exceeded  my  income  by  15 
per  cent.,  I  find  that  by  saving  J  of  my  income  for  the 
succeeding  year,  I  can  supply  the  deficiency  with  interest, 
and  have  $4.60  left.     What  is  my  income  ?    Ans.  $600. 

55.  Find  11  terms  of  a  harmonical  progression,  two  of 
the  terms  being  4  and  8. 

Ans.  8,  4,  2|,  2,  If,  1J,  LI,  1,  |,  .4,  A. 

56.  At  the  breaking  up  of  the  ice  in  a  river,  a  tree  is 
cut  down  by  a  block  of  ice,  which  has  a  surface  of  10000 

23 


354 


MISCELLANEOUS   PROBLEMS.      [ART.  XXIII. 


square  feet,  and  is  1  foot  thick.  How  many  axes,  each 
weighing  10  lb.,  and  moving  with  a  velocity  of  20ft.  per 
second,  would  have  the  same  momentum,  the  velocity  of 
the  ice  being  3-Jft.  per  second,  and  its  specific  gravity  930? 

Ana.  86871. 

57.  Gregory  King,  in  1695,  made  the  following  estimate 
of  the  expense  of  England,  France,  and  Holland,  in  diet  :a 


England. 

France. 

Holland. 

Total. 

In  bread-stuffs  . 

£4300000 

£10600000 

£1400000 

In  meats       .     . 

3300000 

5600000 

800000 

In  butter,   cheese 

2300000 

4200000 

600000 

and  milk   .     . 

In  malt  liquors 

5800000 

100000 

1200000 

In  spirituous  drinks 

1300000 

9000000 

400000 

In  fish,  fowls,  and 

eggs      .... 

1700000 

3900000 

1100000 

In  fruits  and  garden 

produce     .     .     . 

1200000 

3600000 

400000 

In    groceries     and 

sweetmeats    .     . 

1100000 

3000000 

300000 

Total 

58.  Estimating  the  population  of  England,  in  1695,  at 
5 1  million,  France  at  13-J  million,  and  Holland  at  21  mil 
lion,  what  was  the  average  amount  annually  expended  for 
each  article  of  diet  by  each  individual,  in  each   nation  ? 
What  was  the  average  annual  expense  of  each  individual  in 
the  three  nations  ?  2d  Ans.  £3  3s.  4.7d. 

59.  A  man  spends  25  cents  a  day  for  wine  and  cigars ; 
how  much  will  he  lose  by  the  expenditure  in  48  years,  sup 
posing  money  to  be  worth  6  per  cent.  ? 

Ans.  $23427.55. 

60.  A  road  has  been   constructed  through   a   farmer/ s 
land,  taking    li  acres,  worth  $75  per  acre.     In    addition 
to  the  loss  of  his  land,  he  is  obliged  to  expend  $100  in 


Wade. 


§110.]  MISCELLANEOUS   EXAMPLES.  355 

building  a  fence,  which  must  be  renewed  every  12  years. 
What  damages  should  he  receive,  money  being  worth  6  per 
cent,  compound  interest  ?  Ans.  $292.54. 

61.  Find  6  weights  with  which  any  number  of  pounds, 
from  1  to  364,  can  be  weighed. 

62.  "What  is  the  difference  between  the  area  of  a  circle 
whose    circumference   is    157225.ft.,    and   the   area   of  the 
greatest  square  that  can  be  inscribed  in  it  ? 

Ans.  713.5ft. 

63.  What  number  is  that  which  is  divisible  by  11,  but 
if  divided  by  any  number  less  than  11,  leaves  1  remainder  ? 

Ans.  25201. 

64.  The  average  effect  of  a  bushel  of  coals,  weighing 
60  lb.,  when  consumed  by  an  engine  now  working  at  Wheal 
Towan,  in  Cornwall,  is  sufficient  to  raise  70  million  pounds 
one  foot  high.a     How  many  pounds  of  the  same  coal  would 
furnish  sufficient  power  to  raise  a  man,  weighing  150  lb.,  to 
the  summit  of  Mont  Blanc,  an  elevation  of  15680ft.  ? 

Ans. 


65.  The  Menai  bridge  consists  of  a  mass  of  iron  about 
4  million  pounds  in  weight,  suspended  at  an  average  height 
of  170ft.  above  the  sea.a     How  many  bushels  of  coal  would 
have  sufficed  to  raise  it  to  its  present  position  ? 

66.  Estimating  the  quantity  of  granite  in  the  great  pyra 
mid  at  75614816  c.  ft.,  the  specific  gravity  at  2700,  and 
the  average  height  to  which  the  materials  were  raised  at 
125ft.,  how  many  chaldrons  of  coal  would  have  furnished 
the  power  necessary  for  its  erection  ? 

Ans.  632-J-fi|§jj  chaldrons,  a  quantity  consumed  in 
some  foundries  in  a  week. 

67.  The  entire  expense  of  the  Revolutionary  War  was 
estimated   by   the   Register    of   the    Treasury,    in    1790, 

a  Working  Man's  Friend. 


356  MISCELLANEOUS   PROBLEMS.       [ART.  XXIII. 

at  8135193700,  to  meet  which  an  issue  was  made  of 
$359547027T7g  in  continental  money.a  What  was  the  average 
loss  per  cent,  by  the  depreciation  of  the  continental  cur 
rency?  Ans.  62.398+  per  cent. 

68.  Four  men  bought  a  grindstone,  40  inches  in  diame 
ter,  each  contributing   an  equal  amount.     How  much  of 
the  diameter  ought  each  to  grind  away  ? 

Ans.  1st.  5.359in.  ;  2d.  6.3568in.  ; 
3d.  8.2842in.  ;  4th.  20in. 

69.  A.  and  B.  are  on  opposite  sides  of  a  circular  field  that 
is  120  rods  in  diameter,  and  commence  travelling  around  it 
in  the  same  direction.    How  many  times  will  each  go  round 
the  field  before  the  slower  is  overtaken,  A.  going  39  rods 
in  3  minutes,  and  B.  66|  rods  in  5  minutes  ? 

Ans.  A.  19£  times;  B.  20  times. 

70.  A  man  sold  a  horse  for  $65.25,  thereby  gaining  as 
much  per  cent,  as  the  horse  cost  him.     What  did  he  give 
for  the  horse  ?  Ans.  $45. 

71.  Professor  Wheatstone  endeavored  to  determine  the 
velocity  of  electricity  by  using  a  revolving   mirror,   and 
observing  the  relative  position  of  the  images  made  by  the 
sparks  at  the  two  extremities  of  a  long  wire.     What  inter 
val  of  time  would  be  indicated  by  an  angular  deviation  of 
1°  in  the  appearance  of  the  two  sparks,  supposing  the  mirror 
to  make  800  revolutions  in  1  second  ?        Ans.  sec* 


72.  It  has  been  estimated  that  the  average  quantity  of 
air  contaminated  by  respiration,  insensible  perspiration,  and 
lights,  is  4  c.  ft.  per  minute  for  each  individual  ;  the  ave 
rage  amount  cooled  by  the  draught  from  each  door  or  window, 
11  c.  ft.  per  minute  ;  the  number  of  c.  ft.  cooled  by  radia 
tion  through  the  windows,  1|  times  the  number  of  sq.  ft. 
of  the  glass  exposed  to  the  external  air.b  According  to 
this  estimate,  how  much  fresh  air  should  be  supplied  per 

*  Encyclopaedia  Americana.  b  Tredgold. 


§110.]  MISCELLANEOUS   EXAMPLES.  357 

minute,  in  summer,  for  a  hall  containing  2000  people  ?  How 
much  heated  fresh  air,  in  winter,  for  a  church  with  a  congre 
gation  of  600,  there  being  28  windows  and  doors,  and  1000 
sq.  ft.  of  glass  ?  1st  Ans.  8000  c.  ft.  per  min. 

2d  Ans.  4208  c.  ft.  per  min. 

73.  A.  and  B.  own  adjoining  farms.     A.  and  his  family 
do  no  labor  in  winter,  except  to  take  the  necessary  care  of 
his  stock  and  household;  but  the  family  of  B.,  by  shoe- 
making,  braiding  straw,   and   other  similar  employments, 
make  $125.     To  how  much  will  B.'s  winter  labor  amount 
in  30  years,  if  the  proceeds  are  all  invested  at  6  per  cent, 
compound  interest?  Ans.  $9882.27. 

74.  A  farmer  has  a  lane  leading  through  his  pasture,  to 
the  public  highway,  and  instead  of  fencing  in  the  lane,  he 
has  a  gate  at  each  extremity.     If  1£  minutes'  delay  is 
occasioned  by  opening  and  closing  each  gate,  and  if  he  is 
obliged  to  pass  through  the  lane  12  times  a  day,  how  much 
will  he  lose  by  the  gates  in  a  year,  supposing  his  time  to 
be  worth  20  cents  an  hour  ?  Ans.  $36.50. 

75.  A  pump  in  which  the  water  is  to  be  raised  to  the 
height  of  10ft.,  has  a  bore  6in.  in  diameter.     What  should 
be  the  diameter  of  the  bore  of  a  pump,  which  is  to  raise 
the  water  25  feet,  in  order  that  the  two  pumps  may  be 
worked  with  equal  ease  ?  Ans.  3.79in. 

76.  Suppose  a  man  whose  average  weight  is  160  Ibs.,  to 
drink  three  half-pint  cups  of  coffee  per  day  for  40  years,  the 
average  specific  gravity  of  the  coflee  being  1100,  to  how 
many  times  his  own  weight  will  the  whole  amount  ? 

Ans.  24106Jlb.,  or  150.665625  times  his  own  weight. 

77.  It  has  been  stated  by  one  of  the  most  careful  and 
successful  manufacturers,  that  on  substituting,  in  one  of 
his  cotton  mills,  a  better  for  a  poorer  educated  class  of  ope 
ratives,  he  was  able  to  add  12  per  cent,  to  the  speed  of  his 
machinery,  without  any  increase  of  damage  or  danger  from 
the  acceleration.     What   amount  would  be  saved  by  the 


358  MISCELLANEOUS   PROBLEMS.       [ART.  XXHI. 

employment  of  educated  labor,  from  this  source  alone,  in  a 
business  of  $500000  ? 

78.  The  number  of  females  engaged  in  the  various  manu 
factures  of  Massachusetts,  has  been  estimated  at  40000,  and 
their   average   annual  wages   at  $100  each.     The   super 
intendent  of  the  Merrimack  Mills,  in  1841,  estimated  the 
average  wages  of  the  best  educated  operatives,  at  17 1  per 
cent,  above  the  general  average  wages  of  the  mills,  and  the 
average  wages  of  the  least  educated,  at  18  J  per  cent,  below 
the  general  average.     According  to  this  estimate,  how  much 
would  be  gained  by  elevating  the  whole  40000  to  the  highest 
standard,  and  how  much  would  be  lost  by  degrading  them 
to  the  lowest  standard  ? 

79.  The  researches  and  discoveries  of  M.  Meneville,  in 
regard  to  the  fly  which  was  lately  so  destructive  to  the  olive 
in  the  south  of  France,  are  said  to  have  increased  the  value 
of  the   annual   product    of  this  fruit,  $1000000.     What 
amount  of  profit  will  France  derive  in  50  years,  from  the 
education  which  led  to  such  a  discovery,  supposing    the 
entire  increase  of  value  to  be  invested  annually  at  5  per 
cent,  compound  interest  ? 

80.  The  aggregate  quantity  of  water  annually  discharged  by 
the  Mississippi  river,  has  been  estimated  at  14883360636880 
c.  ft.*     To  how  many  cubic  miles  is  this  equivalent  ? 

Ans.  101.1  c.  miles. 

81.  Estimating   the   area  of  the   Mississippi  valley  at 
1400000  sq.  miles,1  and  the  average  depth  of  rain  water 
that  falls  annually  in  the  valley,  at  52  inches,1  how  many 
cubic  feet  of  water  fall  in  the  whole  valley  during  the  year, 
and  what  part  of  it  passes  off  by  evaporation  ? 

Ans.  169128960000000  c.  ft.,  of  which  about  |-J 
passes  off  by  evaporation,  and  /3  is  dis 
charged  by  the  river. 

*  Proceedings  of  Amer.  Association,  1848. 


§110.]  MISCELLANEOUS   EXAMPLES.  359 

82.  The  removal  of  the  forests  from  the  valley  of  the 
Mississippi  has  increased  evaporation  to  such  an  extent, 
that  the  inundations  of  the  river  have  become  much  less 
frequent  and  less  formidable  than  they  were  at  the  first  set 
tlement  of  the  country.*  What  amount  of  water  was  annually 
discharged  by  the  Mississippi,  30  years  ago,  supposing  that 
it  has  since  decreased  25  per  cent.  ? 

Ans.  19844480849173  c.  ft. 

83.  If  23232  c.  in.  of  the  Mississippi  water,  deposit  44 
c.  in.  of  sediment/  how  long  would  it  require  to  deposit  the 
present  delta,  which  is  estimated  to  contain  13600  sq.  m. 
of  surface,  and  to  be  of  the  average  depth  of  -J   mile  ? 

Ans.  14203|  years. 

84.  What  is  the  approximate  weight  of  the  earth,  esti 
mating  its  mean  diameter  at  7912  miles? 

Ans.  13527679878424215242145792  Ib. 

85.  If  12  oxen  eat  3£  acres  of  grass  in  4  weeks,  and  21 
oxen  eat  10  acres  of  the  like  pasture  in  9  weeks,  how  many 
oxen  will  eat  24  acres  in  18  weeks,  the  grass  being  at  first 
equal  on  every  acre,  and  growing  uniformly  ? 

This  example  is  taken  from  Newton's  Universal  Arithmetic. 
It  can  be  solved  most  readily  by  making  three  distinct  questions. 

(1.)  If  12  oxen  eat  l-f  acres  of  grass,  with  the  growth,  in  4  weeks, 
how  many  oxen  will  eat  24  acres,  with  4  weeks'  growth,  in  18 
weeks  ? 

Stating  the  question  by  analysis,  or  by  proportion,       # 
we  obtain  —  — 

4 


(2.)  If  21  oxen  eat  10  acres,  with  the  growth,  in  9  weeks,  how 
many  oxen  will  eat  24  acres,  with.  9  weeks'  growth,  in  18  weeks  ? 

a  Proc.  of  Amer.  Association,  1848. 


MISCELLANEOUS   PROBLEMS.      [ART.  XXIII. 
The  answer,  found  as  before,  is  21  — 


6;ijM05 

Now,  if  19|  oxen  in  18  weeks,  eat  24  acres  with  4  weeks'  growth, 
and  25g  oxen  in  the  same  time,  eat  the  same  number  of  acres 
with  9  weeks'  growth,  5  weeks'  growth  on  24  acres  will  support 
6  oxen  18  weeks.  Then, 

(8.)  If  6  oxen  in  18  weeks  eat  5  weeks'  growth,  how  many  oxen 
in  the  same  time  will  eat  9  weeks'  growth  ? 

The  answer  is  10-.     We  have  already  found  that  25 1        6  — 
oxen  in  18  weeks,  will  eat  24  acres  with  9  weeks'  growth,       "T  7~ 
and  if  we  add  the  number  which  would  eat  the  growth 
of  the  remaining  9  weeks,  we  obtain  86  oxen  for  the  answer  sought. 

86.  If  15  oxen  eat  4£  acres  of  grass  in  2  weeks,  and  24 
oxen  eat  14 J  acres  in  5  weeks,  how  many  oxen  will  eat  48 
acres  in  8  weeks,  the  grass  being  at  first  equal  on  every 
acre,  and  growing  uniformly  ?  Ans.  60  oxen. 

87.  If  11  oxen  eat  24  J  acres  of  grass  in  5  weeks,  and  10 
oxen  eat  26|    acres  in  4  weeks,  how  many  acres  of  similar 
pasture  will  42  oxen  eat  in  7  weeks,  the  grass  growing  uni 
formly?  Ans.  78 g  A. 

88.  What  number  is  that  which  is  169  greater  than  the 
greatest  square  number  below,  and  114  less  than  the  least 
square  number  above  itself?  Ans.  20050. 

89.  A.  and  B.  can  do  T!Q  of  a  piece  of  work  in  a  day,  B. 
and  C.  can  do  J  of  it  in  2{  days,  and  A.  and  C.  can  do  £ 
of  it  in  4y1y  days.     In  what  time  would  each  do  it  alone, 
and  in  what  time  would  it  be  done  if  they  all  worked 
together?  Ans.  A.  in  15  days;  B.  in  30  days; 

C.  in  18  days;  all  together  in  6-f  days. 

90.  There  are  some  monads  not  exceeding  £?^$ui*  in 
diameter,  in  which  6  spots  have  been  observed,  separated 
by  membranous  partitions  not  thicker  than  -^  of  the  diameter 


§110.]  MISCELLANEOUS   EXAMPLES.  361 

of  the  spots.     If  these  estimates  are  correct,  what  is  the 
thickness  of  the  partitions  ?  Ans.  sHgo^o"1- 

91.  What  amount  can  be  recovered  from  the  underwriters, 
upon  the  following  transaction  ? — .£5000  was  insured  upon 
goods  from  New  York  to  London,  the  goods  being  valued 
in  the  invoice,  at  .£7200.     In  a  storm,  a  part  of  this  property 
valued  at  £1107  5s.  was  thrown  overboard,  and  the  remain 
der  was  so  much  damaged,  that  it  sold  for  only  £5407  15s. 
6d.,  whereas  if  it  had  arrived  safe,  it  would    have  sold 
for  £6723  10s. ;  besides  which,  the  owner  of  the  goods  was 
obliged  to  contribute  towards  a  general  average,  at  the  rate 
of  2.225  per  cent,  on  the  invoice  value  of  the  whole. 

Ans.  £1025. 

92.  Exported  45  hogsheads  of  sugar  from  New  York  to 
Amsterdam,  weighing  gross  454cwt.  2qr.  18  lb.,  tare  53  Ib. 
per  hhd.,  which  were  sold  at  12 £  groats  per  lb.,  subject  to  a 
discount  of  5  per  cent.     The  original  cost  and  charges  in 
New  York,  were  $2437.50,  the  amount  of  charges  in  Am 
sterdam,  1149  florins  7  groats,  and  the  agio  of  the  bank  of 
Holland  was  4£  per  cent.     How  much  was  gained  or  lost 
by  the  adventure,  the  net  proceeds  being  remitted  at  the 
exchange  of  40cts.  per  florin  ?      Ans.  $2638.19  gained. 

93.  How  far  can  the  top  of  Bunker  Hill  Monument,  which 
is  282ft.  above  the  level  of  the  sea,  be  seen  from  the  deck  of 
a  vessel,  the  spectator's  eye  being  15ft.  above  the  water  ?a 

Ans.  25.3  miles. 

94.  The  average  power  of  draft  of  a  horse,  moving  5 

a  The  distance  at  which  bodies  may  be  seen,  is  found  by  the  follow 
ing  rule : — 

To  the  earth's  diameter  (41815224  feet),  add  the  height  of  the  eye, 
and  multiply  the  sum  by  the  height  of  the  eye.  The  square  root  of 
the  product  is  the  distance  at  which  an  object  ON  THE  SURFACE  of  the 
earth  or  water  can  be  seen. 

Work  in  the  same  way  with  the  height  of  the  object,  and  the  sum 
of  the  two  results  is  the  distance  at  which  the  object  may  be  seen. 


362  MISCELLANEOUS    PROBLEMS.       [ART.  XXIII. 

miles  per  hour  for  8  hours  a  day,  being  75  lb.,  what  will  be 
the  annual  cost  of  transportation  over  a  road  40  miles 
long,  on  which  the  average  friction  is  Js  of  the  weight, 
estimating  the  amount  transported  at  90000  tons,  and  the 
value  of  a  horse's  labor  at  62 1  cents  a  day  ? 

Ans.  $67200. 

95.  If  the  road,  in  the  preceding  example,  could   be 
improved  by  macadamizing  or  otherwise,  so  that  the  friction 
would  be  reduced  to  fa  of  the  weight,  how  much  might 
profitably  be  expended  in  making  the  improvement,  money 
being  worth  6  per  cent.  ?  Ans.  $653333 -J. 

96.  The  shadow  cast  by  a  Drummond  light,  at  the  dis 
tance  of  80  rods,  was  observed  to  be  of  the  same  intensity 
as  that  cast  by  the  full  moon,  which  was  shining  at  the 
time.     To  how  many  such  lights  was  the  moon's  light 
equivalent,  her  mean  distance  being  240000  miles  ? 

Ans.  921600000000. 

97.  Find  the  least  3  integers,  such  that  §  of  the  first,  T5? 
of  the  second,  and  fs  of  the  third,  shall  be  equal. 

Ans.  140,  147,  150. 

98.  For  the  purchase  of  a  certain  estate,  A.  offers  $150  pre 
mium,  and  $300  rent  per  annum ;  B.  offers  $400  premium, 
and  $250  per  annum ;  C.  offers  $650  premium,  and  $200 
per  annum,  and  D.  offers  $1800  in  ready  money.     Whose 
offer  is  the  best,  and  what  is  the  difference  between  them, 
computing  5  per  cent,  compound  interest  ? 

Partial  Ans.  A.'s  offer  is  the  best. 

99.  Which  is  of  the  greater  value,  the  income  of  an 
estate  of  $500  a  year  for  15  years  to  come,  or  the  reversion 
of  the  same  estate  for  ever,  at  the  expiration  of  the  15 
years,  interest  at  6  per  cent.  ? 

Ans.  The  income  for  15  years. 

100.  If  a  ball  were  put  in  motion  by  a  force  which 


§  110.]  MISCELLANEOUS    EXAMPLES.  363 

would  drive  it  12  miles  the  first  hour,  10  miles  the  second, 
and  so  on  in  geometrical  progression,  what  distance  would 
it  go  in  the  whole  ?  Ans.  72  miles. 

101.  What  is  the  least  number  which,  if  divided  by  2, 
will  leave  1  remainder ;  by  3  will  leave  2 ;  by  4  will  leave 
3 ;  by  5  will  leave  4 ;  by  6  will  leave  5 ;  but  by  7  will 
leave  no  remainder  ?  Ans.  119. 

102.  Required  the  least  three  numbers,  which,  divided 
by  20,  will  leave  19  remainder;  if  divided  by  19  will  leave 
18,  and  so  on,  (always  leaving  one  less  than  the  divisor),  to 
unity.          Ans.  232792559;  465585119;  698377679. 

103.  A  trader  offers  to  receive  a  young  man  as  partner, 
proposing,  if  he  will  advance  $500,  to  allow  him  $200  per 
annum;    if  he   will    advance  $1000,   to    allow   $275    per 
annum ;  and  if  he  advances  $1500,  he  will  allow  $350  per 
annum.     What   per   cent,   is    offered  for  the  use   of  the 
money,  and  how  much  for  the  young  man's  time  ? 

Ans.  15  per  cent.;  $125  per  annum. 

104.  A  shepherd  sold  to  one  man,  half  his  flock  and 
half  a  sheep ;  to  a  second,  half  the  remainder  and  half  a 
sheep ;  and  to  a  third,  half  the  remainder  and  half  a  sheep, 
when  he  had  20  left.     How  many  had  he  at  first  ? 

Ans.  167. 

105.  The  annual  cost  of  transportation  on  a  road  40 
miles  long,  being  estimated   at  $30000    per  mile,   what 
amount  of  saving  can  be  effected  by  expending  $50000  to 
shorten  the  road  3  miles,  and  $2000000  to  reduce  the  fric 
tion  to  £  its  present  amount,  the  annual  cost  of  repairs 
being  the  same  in  both  cases,  and  the  rate  of  interest  being 
5£  per  cent,?  Ans.  $9677272T^. 

106.  A  city  of  50000  inhabitants  is  to  be  supplied  with 
water,  from  a  river  300  feet  below  the  proposed  reservoir. 
Estimating  the  average  daily  consumption  at  10  ale  gallons 


364  MISCELLANEOUS    PROBLEMS.       [ART.  XXIII. 

for  each  individual,  what  must  be  the  power  of  an  engine 
working  12  hours  a  day,  to  raise  the  requisite  supply  ? 

Ans.  48.53  horse  power. 

107.  A  commission  sale  of  60  bags  of  coffee  was  effected 
at  Rotterdam,  at  22  stivers  per  lb.,  with  an  allowance  of 
2  per  cent.,  and  of  1  per  cent,  on  the  remainder;  weight 
gro.  50cwt.  Iqr.  11  lb.,  draft  1   per  cent.,  tare  61b.  per 
bag ;  commission  and  guarantee,  3  per  cent. ;  other  charges, 
269  florins  10  stivers.     Required  the  net  proceeds  of  the 
sale,  a  bill  being  remitted  at  the  exchange  of  39  cents  per 
florin.  Ans.  $2003.58. 

108.  Bought  in  Cadiz  four  chests  of  Peruvian  bark,  at 
25  rials  of  plate  per  lb.,  weighing  gro.  6cwt.  3qr.  16  lb., 
tare  25  lb.  per  chest.     The  export  duty  was  150  rials  vellon 
per  quintal  of  100  lb.,  and  the  amount  of  other  charges  was 
1210  rials  of  plate.    What  was  the  whole  amount  of  the  pur 
chase  ?  Ans.  $1851.40. 

109.  At  a  time  when  bills  upon  the  treasury  bore  at 
Jamaica  a  premium  of  7£  per  cent.,  and  dollars  a  premium 
of  4  per  cent.,  14000  dollars  were  purchased  and  consigned 
to  London,  to  be  disposed  of  there;  they  weighed  1010 lb. 
3oz.  troy,  and  were  sold  to  the  Bank  of  England  at  5s.  3d. 
per  oz.     The  charges  were,  freight  1$  per  cent.;  14  bags, 
at  6d.  each ;  weighing,   2s.  6d. ;  brokerage,   $  per  cent. ; 
commission,  £  per  cent. ;  and  insurance  on  £3000,  at  4  per 
cent.      It  is  required  to  find  the  rate  per  cent,   of  the 
amount  of  the  charges,  estimated  upon  the  cost  of  the  dol 
lars,  exclusive  of  the  premium,  and  to  find  what  would  have 
been  gained  or  lost,  if  in  preference  to  dollars,  their  actual 
cost  had  been  laid  out  in  government  bills. 

Ans.  Value  of  the  dollars,  £3182  5s.  9d. ;  net  pro 
ceeds  of  the  sales,  £2994  3s.  lid;  rate  of  the 
charges,  6J  per  cent.;  gain  by  government 
bills,  £84  9s.  8d. 


§110.]  MISCELLANEOUS   EXAMPLES.  365 

110.  "  One  evening  I  chanced  with  a  tinker  to  sit, 
Whose  tongue  ran  a  great  deal  too  fast  for  his  wit ; 
He  talked  of  his  art  with  abundance  of  mettle, 

So  I  asked  him  to  make  me  a  flat-bottomed  kettle. 
Let  the  top  and  the  bottom  diameters  be 
In  just  such  proportion  as  five  is  to  three  : 
Twelve  inches  the  depth  I  proposed,  and  no  more, 
And  to  hold  in  ale  gallons  seven  less  than  a  score. 
He  promised  to  do  it,  and  straight  to  work  went, 
But  when  he  had  done  it  he  found  it  too  scant. 
He  altered  it  then,  but  too  big  he  had  made  it ; 
And  when  it  held  right,  the  diameters  failed  it. 
Thus  making  it  often  too  big  and  too  little, 
The  tinker  at  last  had  quite  spoiled  his  kettle, 
But  declared  he  would  bring  his  said  promise  to  pass, 
Or  else  that  he'd  spoil  every  ounce  of  his  brass. 
Now  to  keep  him  from  ruin,  I  pray  find  him  out 
The  diameters'  length,  for  he'll  ne'er  do't  without." 

Ans.  24.4in.;  14.64in. 

111.  Two  vessels  are  30  miles  apart,  and  are  sailing,  the 
first  with,  and  the  second  against  a  current  of  2^  miles  per 
hour.    In  still  water,  each  would  sail  7  miles  per  hour.     In 
what  time  will  they  meet,  and  what  will  be  the  distance  of 
each  from  its  present  position  ? 

112.  At  what  time  between  half  past  7  and  8  o'clock, 
are  the  hour  and  minute  hands  exactly  13  minutes  apart  ? 

Ans.  52T4rm.  past  7. 

113.  The  annual  loss  in  the  United  States,  in  consequence 
of  intemperance,  has  been  estimated  at  $108000000.a     If 
this  amount  were  saved,  how  much  would  be  left  after  pur 
chasing  4000000  sheep  at  $2.50,  400000  head  of  cattle  at 
$25,  200000  cows  at  $20,  40000  horses  at  $100,  500000 
suits  of  men's  clothes  at  $20, 1000000  suits  of  boys'  clothes 

a  Amer.  Temp.  Soc.  Reports. 


366  MISCELLANEOUS   PROBLEMS.       [ART.  XXIII. 

at  $10,  500000  suits  of  womens'  clothes  at  $10,  1000000 
suits  of  girls'  clothes  at  $3,  1200000  bbl.  flour  at  $5, 
800000  bbl.  beef  at  $10,  800000  bbl.  pork  at  $12.50, 
SOOOOOObu.  corn  at  $.50,  2000000bu.  potatoes  at  $.25, 
10000000  Ib.  sugar  at  $.10,  400000  Ib.  rice  at  Sets., 
2000000gal.  molasses  at  40cts.,  building  1000  churches  at 
$5000,  8000  school  houses  at  $500,  and  supporting  50000 
families  at  $300?  Ana.  $180000. 

114.  What  is  the  difference  between  10000  years  according 
to  our  present  calendar,  and  10000  solar  years,  estimating 
the  solar  year  at  365d.  5h.  48m.  49£sec.  ?a 

Ans.  2d.  14h.  30m. 

115.  In  1840,  Prof.  Bessel,  from  the  corrected  parallax 
of  the  star  61  Cygni,  estimated  its  distance  from  the  earth 
at  592200  times  the  mean  distance  of  the  sun.     According 
to  this  estimate,  how  long  would  the  star's  light  be  in  reach 
ing  us?  Ans.  3391dy.  9h.  13m.  45s. 

116.  A  micrometer  screw  is  made  with  threads  -^  of  an 
inch  apart,  and  an  index  is  affixed  to  the  head,  which  points 
to  the  degrees  on  a  graduated  circle.     What  thickness  would 
be  measured  by  turning  the  index  1°  15'  ? 


117.  A.  and  B.  are  carrying  a  weight  of  150  Ib.,  which 
is  suspended  on  a  pole,  at  the  distance  of  2ft.  6in.  from  A., 
and  1ft.  9in.  from  B.     What  amount  does  each  one  support  ? 

Ans.  A.  61  Ib.  12TV>z.;  B.  88  Ib.  3}foz. 

118.  Archimedes  boasted,  that  if  he  had  a  place  to  stand, 
he  could  move  the  world.     If  he  weighed  150  Ib.,  how  far 
would  he  be  obliged  to  move,  in  order  to  move  the  earth  1 
inch?     [See  Ex.  84.] 

Ans.   1423366990574938472.448  miles. 

119.  At  Bilin,  in  Germany,  is  a  bed  of  tripoli,  composed 

a  Carpenter. 


§110.]  MISCELLANEOUS   EXAMPLES.  367 

almost  entirely  of  the  sheaths  of  a  kind  of  animalcule,  the 
length  of  each  sheath  being  about  g^  of  an  inch.a  How 
many  such  sheaths  would  there  be  in  a  cubic  inch  ? 

Ans.  42875000000. 

120.  A  grain  of  copper  dissolved  in  nitric  acid,  and  mixed 
with  three  pints  of  water,  (wine  measure,)  gives  a  blue  color 
to  the  whole.     What  is  the  weight  of  the  quantity  contained 
in  Ttfy  Jooo  °f  a  cu^c  inch>  which  is  sufficiently  large  to 
be  visible  to  the  naked  eye  ?  a 

A™.  HffgaVuuu  of  a  grain. 

121.  In  the  manufacture  of  embroidery  wire,  a  cylinder 
of  silver,  weighing  360oz.,  is  covered  with  2oz.   of  gold. 
This  is  drawn  into  wire,  of  which  4000ft.  weigh  loz.     A 
foot  of  this  wire  may  be  divided  into  1200  parts  visible  to 
the  naked  eye.     What  would  be  the  weight  of  a  particle 
of  the  gold  upon  this  wire,  that  would  be  visible  under  a 
microscope  magnifying  500  times  each  way? 

Ans. 


122.  The  squares  of  the  times  of  revolutions  of  the 
planets  around  the  sun,  are  proportioned  to  the  cubes  of 
their  mean  distances.     The  mean  distance  of  the  earth  from 
the  sun,  is  95  million  miles,  and  it  revolves  round  the  sun 
in  365d.  5h.  48m.  50s.     What  is  the  mean  distance  of 
Yenus,  her  time  of  revolution  being  224d.  16f  h.  ?b 

Ans.  69  million  miles,  nearly. 

123.  What  do  I  gain  per  cent,  by  purchasing  goods  at  8 
months'  credit,  and  selling  them  immediately  for  cash,  at 
the  same  price,  money  being  worth  6  per  cent,  per  annum  ? 

Ans.  3-}-]-  per  cent. 

124.  A.  and  B.   enter  into   partnership,  A.  advancing 
$4800  to  carry  on  the  business.     B.  has  no  money,  but- 
being  thoroughly  acquainted  with  the  trade,  agrees  to  be 
manager.  B.'s  annual  salary,  before  he  engaged  in  the  copart- 

*  Carpenter.  b  Nesbit. 


368  MISCELLANEOUS   PROBLEMS.       [ART.  XXIII. 

nership,  was  $250.  By  the  terms  of  the  contract,  A.  is  to 
be  allowed  7  per  cent,  for  the  interest  of  his  money  and  the 
risk  of  the  business,  and  the  net  profits,  after  making  this 
deduction,  are  to  be  divided  in  the  proportion  which  6  per 
cent,  on  A.'s  capital  bears  to  B.'s  former  salary.  The  profits 
at  the  end  of  the  year,  amounted  to  $5000.  How  should 
they  be  divided?  '  Ans.  A.  $2832.71;  B.  $2167.29. 

125.  What  quantity  of  each  of  the  following  ingredients, 
would  be  required  to  make  a  ton  of  flint  glass,  estimating 
the  waste  in  melting  at  2  per  cent.  ?     White  sand,  9  parts; 
red  lead,  or  litharge,  6.5  parts ;  pearlash  with  a  little  nitre, 
4.5  parts.*  Ans.  Sand  1028.6  Ib. ;  litharge  742.9  Ib. ; 

pearlash  514.3  Ib. 

126.  The  dust  of  the  puff  ball  consists  of  seeds,  which 
vary  from  ^fav  to  *ffi™  of  an  inch  in  diameter.     How 
many  seeds,  of  the  average  diameter  of  ^i^in.,  would 
there  be  in  1 J  cubic  inches  ?        Ans.  234375000000000. 

127.  Estimating   the    entire   wealth    of    the   world   at 
$1407374883553.28,  in  what  time  would  1  cent  absorb  all 
the  property  on  the  globe,  if  it  were  placed  at  compound 
interest,  so  as  to  double  every  11  years  ? 

Ans.  517  years. 

128.  A  merchant  failing  in  business,  offers  to  settle  with 
his  creditors  by  paying  them  80  cents  on  a  dollar  in  6 
months,  without  interest,  or  to  give  security  for  the  pay 
ment  of  the  whole  in  4£  years,  without  interest.     Which 
proposal  is  the  more  advantageous,  supposing  that  the  com 
position    first   proposed  can   be   invested    at   6  per  cent, 
compound  interest  ? 

129.  Bought  flour  for  $3.50  per  barrel,  on  6  months. 
At  what  price  must  it  be  sold  to  gain  20  per  cent.,  and  allow 
3  months'  credit,  money  being  worth  8  per  cent,  per  annum  ? 

Ans.  $4.11ff. 

»  Parnell. 


§110.]  MISCELLANEOUS   EXAMPLES.  369 

130.  A.  has  TOcwt.  of  sugar,  which  he  would  sell  for 
$7  per  cwt.  for  cash,  but  in  barter  he  values  it  at  $8.50  per 
cwt.     B.  has  wheat  worth  in  ready  money  $1.12  per  bu., 
which  he  wishes  to  exchange  with  A.    At  what  price  should 
it  be  valued  in  barter  ?  Ans.  $1.36. 

131.  Insured  150  hogsheads  of  sugar,  valued  at  $125  per 
hogshead,  from  Jamaica  to  Boston,  at  8  per  cent.,  to  return 
4  per  cent,  for  convoy  and  arrival  without  loss.     Of  the 
quantity  insured,  30hhds.  were  shipped  on  board  a  vessel 
that  was  totally  lost,  64hhds.  arrived  safe  by  a  vessel  that 
sailed   with   convoy   the   whole    of  her   voyage,   45hhds. 
arrived  with  convoy,  but  in  consequence  of  a  mast  being 
cut  away,  and  some  goods  being  thrown  overboard  to  lighten 
the  vessel  in  a  storm,  the  merchandise  shipped  was  obliged 
to  contribute  to  the  loss,  at  the  rate  of  2  per  cent,  on  its 
value,  and  the  rest  of  the  sugar  was  not  shipped.     Required 
the  amount  to  be  recovered  from  the  underwriters,  and  the 
amount  of  return  premium  for  short  interest,  and  for  convoy 
and  arrival.1  Ans.  Underwriters  to  pay  $3862.50. 

Return  premium,         $419.00. 

132.  What  is  the  product  of  $18.75  by  $.06*  ? 

133.  James  Davis,  of  Waverly,  Ross  county,  Ohio,  culti 
vated,  in  the  year  1849, 1800  acres  exclusively  in  Indian  corn. 
With  the  crop  he  filled  a  corn-crib  3  miles  long,  10ft.  high, 
and  6ft.  wide.b     What  was  the  average  yield  per  acre,  and 
how  many  bushels  would  the  whole  make,  when  shelled  ? 

Ans.  422. 4bu.  in  the  ear,  per  acre. 
396000bu.,  when  shelled. 

1  The  custom  varies  as  to  the  amount  of  return  premium,  when  part 
of  the  risk  is  avoided.  Some  companies  reserve  i  per  cent,  on  the 
amount  insured  ;  some  reserve  10  per  cent,  of  the  premium  paid  in  ; 
on  "open  policies,"  no  premium  is  usually  paid,  except  for  the 
amount  actually  at  risk.  In  the  present  instance,  10  per  cent,  of  the 
premium  paid  on  the  molasses  which  was  not  shipped,  is  supposed  to 
be  reserved  by  the  company. 

b  Cincinnati  Gazette. 

24 


370  MISCELLANEOUS   PROBLEMS.       [ART.  XXIII. 

134.  Sysla,  the  reputed  inventor  of  the  game  of  chess, 
is  said  to  have  asked  as  a  reward,  one  grain  of  wheat  for 
the  first  square  on  the  chess-board,  two  for  the  second,  and 
so  on  in  geometrical  progression.     What  would  have  been 
the  amount  of  his  reward,  there  being  64  squares  on  the 
board,  and  9200  grains  of  wheat  in  a  pint  ?     What  would 
be  the  height  of  a  cubical  bin  that  would  contain  it,  sup 
posing  the  base  to  be  10  miles  square  ? 

Ans.  18446744073709551615  grains; 

31329388712142.58  bushels;  height  of 
bin,  2.65  miles. 

135.  If  the  cost  of  a  railroad  is  $40000  per  mile,  and 
the  annual  repairs  and  expenses  are  estimated  at  $2500  per 
mile,  how  much  may  be  profitably  expended  at  the  outset, 
in  order  to  shorten  the  proposed  route  2J  miles,  provided 
the  stock  pays  an  annual  dividend  of  8  per  cent.  ? 

Ans.  $178125. 

136.  What  is  the  weight  of  a  round  iron  rod,  10  feet 
long,  and  4  inches  in  diameter?         Ans.   538 Ib.  lOfoz. 

137.  A  mule  and  an  ass  travelling  together,  the    ass 
began  to  complain  that  her  burthen  was  too  heavy.    "  Lazy 
animal,"  said  the  mule,  "you  have  little  reason  to  com 
plain  ;  for  if  I  take  two  of  your  bags,  I  shall  have  three 
times  as  many  as  you,  but  if  I  give  you  two  of  mine,  we 
shall  have  only  an  equal  number."     With  how  many  bags 
was  each  loaded? 

138.  How  would  you  distribute  among  3   persons   21 
bags,  7  of  which  are  full  of  corn,  7  half  full,  and  7  empty, 
so  as  to  give  to  each  the  same  quantity  of  corn  and  the 
same  number  of  bags  ? 

This  question  admits  of  two  solutions. 

139.  What  is  the  7th  root  of  63  ?      Ans.  1.80737  +  . 

140.  A  farmer  has  a  stack  of  hay,  from  which  he  sells  a 


§110.]  MISCELLANEOUS   EXAMPLES.  371 

quantity  which  is  to  the  quantity  remaining,  as  4  to  5.  He 
then  uses  15  loads,  and  finds  that  the  quantity  left  is  to 
the  quantity  sold,  as  1  to  2.  Kequired  the  number  of  loads 
at  first  in  the  stack.  Am.  45  loads. 

141.  At  what  points  would  the  numbering  on  Fahren 
heit's  and  on  Delisle's  thermometer-scale  be  the  same,  the 
only  difference  being  in  the  sign  ?a 

Aw.   +  96T4T°Fah.;   -96T4T°Del. 

142.  If  the  temperature  below  0°  on  the  Russian  scale 
is  marked  +,  and  the  temperature  above  0°  is  marked  — ,b 
at  what  points  will  the  numbering  be  the  same,  on  the 
Russian  and  Centigrade  scales?  Am.  60°. 

143.  The  prime  cost  of  109cwt.  3qr.  18  Ib.  of  sugar  was 
$769.375;    the  freight,    insurance,    and    other    expenses, 
amounted  to  $92.31.     What  did  it  cost  per  cwt.,  and  at 
what  price  must  it  be  sold  per  Ib.  to  gain  20  per  cent.,  sup 
posing  4  per  cent,  to  be  lost  by  overweight  in  retailing  ? 

Ans.  $7  per  cwt. ;  8f  cts.  per  Ib. 

144.  A  merchant  receives  an  invoice  of  goods,  which  will 
probably  sell  for  $16000,  by  which  he  will  realize  a  profit 
of  33  J  per  cent,  above  the  prime  cost;  but  preferring  to 
sacrifice  a  portion  of  his  profit,  rather  than  to  risk  the  entire 
loss   of  the  goods,  he  determines  to  insure  them.     The 
premium  is  3J  per  cent.,  the  policy  $2.00,  the  agent's  com 
mission   J  per  cent.,  and  if  there  is  any  loss,  the  broker 
will  charge  1  per  cent,  for  procuring  a  settlement  with  the 
underwriters.     What  amount  should  the  merchant  insure, 

*  The  pupil  should  observe  that  Reaumur's  and  the  Centigrade 
scales  correspond  only  at  0°,  but  all  the  other  scales  have  the  same 
numbering  at  two  points  of  equal  temperature,  differing  at  one  of  the 
points  only  by  the  sign. 

b  The  signs  are  sometimes  applied  in  this  way  on  the  Russian  scale, 
but  the  notation  given  on  p.  197,  appears  more  philosophical,  on  accoun* 
of  its  uniformity. 


372  MISCELLANEOUS   PROBLEMS.       [ART.  XXIII 

in  order  to  recover  the  prime  cost  of  the  goods,  and  all 
expenses  attending  the  insurance,  in  case  of  a  total  loss  ? 

Am.  $12633.68. 

145.  A  trader  wishes  to  sell  his  merchandise  at  whole- 
gale,  so  as  to  make  a  profit  of  20  per  cent.,  after  deducting 
30°|0  from  the  retail  price.     He  therefore  adds  50  per  cent, 
to  the  cost  of  the  articles  in  cash,  and  deducts  30  per  cent, 
from  the  amount,  for  the  wholesale  price,  allowing  a  credit 
of  8  months.     Computing  interest  at  6  per  cent,  per  annum, 
what  is  his  actual  loss  on  all  sales  at  this  rate,  supposing 
that  his  expenses  and  bad  debts  amount  to  15  per  cent, 
of  his  total  sales?  Ans.  14|f  J  per  cent. 

146.  What  percentage  should  have  been  added  to  the 
prime  cost  of  the  articles  in  the  foregoing  example,  to  yield 
a  net  profit  of  10  per  cent,  after  making  the  discount  pro 
posed?  Ans.  92T3T\  per  cent. 

147.  A  merchant  received  on  consignment  three  bales  of 
sheeting,  marked  A.,  B.,  and  C.     A.  contained  420  yards 
of  a  quality  15  per  cent,  better  than  B.,  B.  contained  380 
yards  of  a  quality  10  per  cent,  poorer  than  C.,  and  C.  con 
tained  450  yards.     The  whole  were  sold  together  at  12  J 
cents  per  yard ;  how  much  should  be  credited  to  each,  after 
deducting  2£  per  cent,  commission? 

Ans.  A.  $53.98;  B.  $42.47;  C.  $55.89. 

148.  An  estate  was  offered  for  sale  for  $12000,  but  the 
price  appearing  too  high,  the  tenant  took  a  lease  for  25  years 
at  $800  per  annum.     How  much  did  he  gain  or  lose,  esti 
mating  compound  interest  at  6  per  cent.  ? 

Ans.  $1022.67. 

149.  If  32  oxen  in  3  weeks  consume  all  the  grass  on 
23  acres  of  pasture,  and  if  22  oxen  in  5  weeks  consume  the 
grass  on  25  acres,  in  what  time  will  88  oxen  consume  115 
acres  of  similar  pasturage,  the  grass  growing  uniformly  ? 

Ans.  5.88  weeks. 


§110.]  MISCELLANEOUS   EXAMPLES.  373 

150.  The  steam  power  at   present   employed  in  Great 
Britain  and  Ireland,  is  estimated  as  equivalent  to  the  power 
of  8000000  men.a     If  the  power  of  a  horse  is  5  times  as 
great  as  that  of  a  man,  and  if  a  horse  requires  8  times  the 
quantity  of  soil  for  producing  food  that  a  human  being 
does,  what  population  could  be  supported  by  the  additional 
food  which  would  be  required,  if  horse  power  were  substi 
tuted  for  steam  ?  Ans.  12800000. 

151.  The  hydraulic  press  of  Bramah  can,  by  the  exertion  of 
a  single  man,  produce  a  pressure  of  1500  atmospheres.b 
Estimating  the  mean  height  of  the  mercury  in  the  barometer 
at  29.53  inches,  what  would  be  the   pressure  exerted  on  a 
surface  2ft.  3iu.  long,  and  1ft.  9in.  wide  ? 

Ans.  5502T.  5cwt.  Iqr.  15  Ib.  12oz. 

152.  A  species  of  lace  is  made  by  covering  an  inclined 
flat  surface  with  a  paste  made  of  leaves,  and  drawing  with 
a  earners  hair  pencil,  the  pattern  which  is  to  be  left  open. 
A  number  of  caterpillars,  of  a  species  which  spins  a  strong 
web,  are  then  placed  at  the   bottom,  and  they  commence 
eating  and  spinning  their  way  to  the  top,  carefully  avoiding 
every  part  touched  by  the  oil,  but  devouring  every  other 
part  of  the  paste.b     How  many  square  yards  of  the  lace  thus 
made,  would   there    be  in    1  Ib.  avoirdupois,  the  average 
weight  being  4£  grains  troy  per  sq.  yd.  ? 

Ans.   1615T53  sq.  yd. 

153.  Estimating  the  weight  of  a  globe  of  air  1ft.   in 
diameter,  at  ^V  Ib.  avoirdupois,  and  the  weight  of  an  equal 
globe  of  hydrogen,  allowing  for  impurities,  at  J  as  much, 
what  weight  would   be   sustained   by  a  balloon  18ft.   in 
diameter,  if  fully  inflated  with  hydrogen  gas  ? 

Ans.  194.4  Ib. 

154.  Having  observed  that  the  shadow  of  a  cloud  is  15 
seconds  in  passing  from  one  point  to  another,  I  measure 

a  Chambers'  Mechanics.  b  Babbage. 


374  MISCELLANEOUS   PROBLEMS.       [ART.  XXIII. 

the  distance  between  the  two  points,  and  find  that  it  is  31  £ 
rods.     What  is  the  velocity  of  the  wind  per  hour  ? 

Ans.  23m.  200r. 

155.  A.,  B.  and  C.  have  a  loaf  of  sugar,  weighing  48  lb., 
which  they  wish  to  divide  equally  between  them,  but  having 
only  a  4  lb.  weight  and  a  7  Ib.  weight,  it  is  required  to  find 
how  the  division  can  be  made. 

156.  A  hundred  hurdles  may  be  so  placed,  as  to  enclose 
200  sheep,  and  with  2  more  the  fold  may  be  so  made  as 
to  hold  400.     How  can  this  be  done  ? 


157.  The  animalcules  of  iron  ochre,  are  about  T2J^^  of 
an  inch  in  diameter.     How  many  such  animalcules  would 
occupy  1  c.  ft.  ?  Ans.  2985984000000000. 

158.  An  ounce  of  gold  forms  a  cube  about  -f2  of  an  inch 
thick,  but  by  hammering,  it  may  be  extended  so  as  to  cover 
a  surface  of  146  sq.  ft.     How  many  leaves  formed  in  this 
manner,  would  equal  in  thickness  a  leaf  of  writing  paper 
T-J.3  of  an  inch  thick  ?  Ans.  1938. 

159.  Bought  wheat  for  cash,  at  8.90,  at  8.95,  and  at 
$1.10  per  bushel.     In  what  proportions  may  the  three  kinds 
be  mixed,  so  as  to  gain  20  per  cent,  by  selling  at  81.25  per 
bushel,,  on  6  months'  credit,  money  being  worth  7  per  cent. 
per  annum?  Ans.  11.62bu.  at  8.90; 

11.62bu.  at  8.95;  20.13bu.  at  81.10. 

160.  A  farmer  has  paid  in  cash  84000  for  the  lease  of  a 
farm  for  8  years.     If  money  is  worth  5  per  cent,  compound 
interest,  what  income  ought  he  to  receive  from  the  farm  each 
year,  in  order  to  recover  his  outlay,  and  lay  up  8200  a  year, 
his  family  expenses  being  $350  ?  Ans.  81168.89. 

161.  A  family  of  10  persons  hired  a  house  for  6  months, 
at  a  rent  of  8780  per  annum.     At  the  expiration  of  14 
weeks  they  received  4  new  boarders,  and  at  the  expiration 
of  every  3  weeks,  during  the  remainder  of  the  term,  they 


§110.]  MISCELLANEOUS   EXAMPLES.  375 

received  4  more.     How  much  of  the  rent  should  be  paid 
by  one  of  each  class  ?  Ans.  1st  class  S30.49T§^T; 

2d  class  $9.49TgJT;  3d  class  S6.271 


4th  class 


$3.77 


143 


5th  class  $1.73TV 


162.  Estimate  of  the  cost  of  manufacturing  pins,  "Ele 
vens,"  of  which  5546  weigh  1  Ib.  :a — 


Time  of 

Cost  of  ma- 

!ij? 

NAME    OF    THE   PROCESS. 

Hands. 

making  lib 

kin;  1  Ib. 

Hand  earns 

^.g.0— 

of  pins. 

gj=-i?.2  b 

£S-§s8. 

Hours 

Pen  t 

1.  Drawing  wire  

Man 

.3636 

1.2500 

3    3 

225 

2.  Straightening  wire  j 

Woman 
Girl 

.3000 
.3000 

.2840 
.1420 

1     0 
0    6 

51 
26 

3.  Pointing  
4.  Twisting     and    cutting  ( 
the  heads  ) 

Man 
Boy 
Man 

.3000 
.0400 
0400 

1.7750 
.0147 

5     3 

0    4i 

319 
3 

6.  Tinning,  or  Whitening.,  j 
7.  Papering  

Man 

Woman 
Wom;m 

.1071 
.1071 
2.1314 

.6666 
.3333 
3.1073 

(5     0 
3    0 
1    6 

901 
121 
60     1 
576     ' 

Estimating  the  premium  on  sterling  money  at  9|  per 
cent.,  what  is  the  cost  of  each  pin,  in  our  currency  ?  At 
what  price  should  the  pins  be  sold  per  ounce,  after  paying 
15  per  cent,  profit  to  the  manufacturer,  30  per  cent,  ad 
valorem  for  duties,  18  per  cent,  to  the  American  importer, 
and  25  per  cent,  to  the  retailer,  allowing  10  per  cent,  of 
the  retail  price  for  loss  of  interest  ? 

An8'   eioiiSltro-  of  a  cent;  3.9837+  cents. 

163.  If  a  powerful  Drummond  light,  at  the  distance  of 
30  feet,  casts  a  shadow  of  the  same  intensity  as  that  cast 
by  the  sun,  to  how  many  such  lights  is  the  sun's  light 
equivalent,  estimating  the  distance  of  the  sun  at  95  million 
miks  ?  Ans.  279558400000000000000. 

164.  It  has  been  supposed  that  10  turns  of  Babbage's 
calculating  machine  may  be  made  in  a  minute/     At  this 
rate,  how  many  places  of  figures  would  the  machine  reach 


876  MISCELLANEOUS    PROBLEMS.        [ART.  XXIII. 

in  a  million  centuries,  supposing  it  to  be  so  regulated  as  to 
commence  with  1,  and  give  all  the  following  numbers  in 
their  natural  order?  Ans.  15  places. 

165.  It  has  been  estimated*  that  a  man  in  a  properly 
ventilated  room  can  work  12  hours  a  day,  with  no  greater 
inconvenience  than  would  be  occasioned  by  10  hours'  work 
in  a  room  badly  ventilated,  and  that  where  there  is  proper 
ventilation,  a  man  may  gain  10  years'  good  labor  on  account 
of  unimpaired  health.     According  to  this  estimate,  what  is 
the  loss,  in  30  years,  to  each  individual  in  a  badly  venti 
lated  workshop,  valuing  the  labor  at  1 0  cents  per  hour  ? 

Ans.  $5008. 

166.  In  the  town  of  Bury,  England,  with  an  estimated 
population  of  25000,  the  expenditure  for  beer  and  spirits, 
in  the  year  1836,  was  estimated  at  £54190.a     If  this  sum 
was  24  per  cent,  of  the  entire  loss,  resulting  from  the  waste 
of  money,   ill  health,  loss  of  labor,   and  the  other  evils 
attendant  upon  intoxication,   what  was  the  average    loss 
from  intemperance,  for  each  man,  woman,  and  child,  in  the 
place,  estimating  the  pound  sterling  at  $4.80? 

Ans.  $43.352. 

111.    TABLE  or  PRIME  AND  COMPOSITE  NUMBERS. 

The  following  table  contains  all  the  prime  numbers,  and 
the  factors  of  all  odd  composite  numbers,  below  12700,  the 
prime  numbers  being  indicated  by  a  dash. 

For  numbers  below  1000,  all  the  factors  are  given.  For  the 
odd  numbers  above  1000,  one  or  more  factors  will  be  found 
in  the  table,  which  will  reduce  each  number  to  a  prime,  or 
to  some  number  less  than  1000,  and  under  the  latter  num 
ber  the  remaining  factors  may  be  found. 

The  hundreds  are  placed  at  the  head  of  the  table,  and 
the  tens  and  units  at  the  left  hand. 

a  British  Sanitary  Reports. 


§111.]      TABLE  OF  PRIME  AND  COMPOSITE  NUMBERS. 


I        0 

fc~ 

01    — 

m  — 

,03   - 

j04      2' 

1 

23.52 
2.3.17 
2».13 

I        ~ 

3 

4 

5 

22.5 
3.167 
2.25 

1        6 
23.3.5a 

2.7.43 
32.67 
22.151 

7 
2*.5».7 

2.33.13 
19.37 

26.11 

8 

25.5^ 
32.89 
2.401 
11.73 
22.3.67 

5.7.2.1 
2.13.31 
3.269 
23.101 

9 

23.52 

3.ir 

2.10 

7.2^ 

23.3.T 

5.4 
2.10: 
32.2.' 
24.13 
11.1! 

22.3.52 
7.4: 
2.15 
3.101 
2«.19 

24.52 

2.3.67 
13.31 
22.101 

2a.3a.5a 
17.53 
2.11.41 
3.7.43 

23.1.I3 

23.7.9 

05    — 
00    2.3 

E    — 
23 
3a 

3.5.7 

2.53 

22.33 

5.<> 
2.32.1- 

3«.5 

2.7.29 
11.37 
23.3.17 

5.101 
2.11.23 
3.132 
22.127 

5.112 

2.3.101 

2s.n 
3.7.29 

3.5.47 
2.353 
7.101 
22.3.59 

5.181 

2.3.1.31 

22.7.1 
3.103 

32.10J 

10    2.5 
11    — 
12  22.3 
13   — 
14    2.7 

2.5.11 
3.37 

2.3.19 

2.3.5.- 

2.5.31 
23.3.K 

2.5.41 
3.137 
22.10! 
7.5J 
2.32.23 

2.3.5.17 

7.7: 

29 
33.1i 

2.2r)7 

2.5.61 
13.47 
22.32.17 

2.307 

2.5.71 
32.79 
23.89 
23.31 
2.3.7.17 

2.3«.5 

22.7.23 
3.271 
2.11.37 

2.5.7.13 

22.5: 
3.71 
2.MT 

2*.3  19 
11.83 
2.4^7 

2.15- 

115    3.5 
ilfi     2* 
17    — 
18  2.33 
19   — 

5.23 
22.21 
32.13 
2.59 
7.17 

5.43 

23.33 
7.31 
2.10.; 
3.73 

32.5.~ 
22.7! 

5.83 
25.13 
3.130 
2.11.19 

5.103 
22.3.43 
11.47 
2.7.37 
3.173 

3.5.41 
23.7.11 

5.11.13 
22.179 

3.23: 

2.359 

5.163 
24.3.17 
19.43 
2.40< 
32.7.13 

3.5.61 
22.229 
7.131 
2.33.17 

2.3.53 
11.2! 

2.3.103 

;20!  23.5 
21     3.7 
9-2  2.11 
0.3   

124  23.3 

25     5' 
20  2.13 
27      3» 

28  22.7 
29    — 

30  2.3.5 
31    — 
32      2s 
33  3.11 
34  2.17 

23.3.5 

112 

2.61 
3.41 
22.31 

22.5.11 
13.17 
2.3.3? 

2».7 

26  5 
3J07 
2.7.23 
17.19 
22.3< 

22.3.5.7 

2.211 
32.47 
23.53 

23.5.13 

22.5.31 
33.23 
2.311 
7.89 
2*.3.13 

24.32.5 
7.103 
2.192 
3.241 
22.181 

22.5.41 

23.5.23 
3.307 
2.^61 
13.71 
J2.3.7.11 

2.32.29 

2.3.137 

22.131 

23.103 

53 

2.32.7 

2' 
3.43 

2.5.13 

22.3.11 
7.19 

2.67 

32.52 
2.113 

22.3.19 

2  5  2[ 
3.7.n 
23.29 

2.32.13 

52.13 
2.163 
3.109 
23.41 
7.47 

2.3.5.11 

22.83 
32.37 
2.167 

5.67 
24.3.7 

2.132 
3.113 

52.17 
2.3.71 
7.61 
22.107 
3.11.13 

2.5.43 
24.33 
2.7.31 

3.5.29 
22.109 
19.23 
2.3.73 

3.52.7 
2.263 
17.31 
24.3.11 
232 

2.5.53 
32.59 
22.7.19 
13.41 
2.3.89 

5* 
2.313 
3.11.19 
22.157 
17.37 

2.32.5.7 

23.79 
3.211 
2.317 

52.29 
2.3.112 

3.52.11 

2.7.59 

52.37 
2.4(i3 
32.103 

25.21 

2.3.5.31 
72.19 
22.233 
3.311 
2.467 

23.7.13 

36 

2.5.73 
17.43 
22.3.61 

2.367 

22.32.23 

2.5.83 
3.277 
28.13 
72.17 
2.3.139 

35     5.7 
3(i  22.32 
37    — 
38  2.19 
3<t  3.13 

33.5 
23.17 

2.3.23 

5.47 
22.59 
3.79 
2.7.17 

5.107 
23.67 
3.179 
2.269 
72.11 

5.127 
22.3.53 
72.13 
2.11.29 
32.71 

3.5.72 
25.23 
11.67 
2.32.41 

5.167 

22.1L1<: 

33.31 
2.419 

5.11.17 

23.32.13 

2^67 
3.313 

40  23.5 
41    — 
42  2.3.7 
13    — 
44  22.11 

22.5.7 
3.47 
2.71 
11.13 
24.32 

24.3.5 

2.112 

3» 

22.61 

22.5.17 
11.31 
2.32.19 
73 
23.43 

23.5.11 
32.72 
2.13.17 

22.3.37 

22.33.5 

2.271 
3.181 
25.17 

2'.5 

22.5.37 
3.13.19 
2.7.53 

23.3.31 

23.3.5.7 
292 
2.421 
3.281 
22.211 

22.5.47 

2.3.107 
22.7.23 

2.3.157 
23.41 
24.59 

45  3^.5 
46  2.23 

47 

5.29 
2.73 
3.72 
22.37 

5.72 
2.3.41 
13.19 
23.31 
3.83 

3.5.23 
2.173 

5.89 
2.223 
3.149 
2«.7 

5.109 
2.3.7.13 

22.137 
32.61 

3.5.43 
2.17.19 

5.149 
2.373 
32.83 
22.11.17 
7.107 

5.132 
2.32.47 

7.112 

2*.53 
3.283 

33.5.7 
2.11.43 

22.3.79 
13.73 

48  24.3 
49     72 

22.3.29 

23.3« 
11.59 

378 


MISCELLANEOUS    PROBLEMS.  [ART.  XXIII. 


•" 

1 

2      |        3 

4 

5 

6 

8            9 

2.53.17  2.53.19 
23.37      3.317 
22.3.7r  23.7.17 

2.7.61   2.32.53 

!50  2.5« 
51  3.17 
5-2  2^.13 
,53  
J54  2.3» 

2.3.5' 

23.19 
33.17 
2.7.11 

2.53 

2.53.7 
33.13 

26.11 

2.3.59 

2.33.53 
11.41 
23.113 
3.151 

2.227 

2.53.11     2.53.13 
19.29       3.7.31 
23.3.23      23.163 

7  ng     

2.277     2.3.109 

2.3.5* 

23.7.9 
11.23 
2.127 

2«.47 
3251 
2.13.29 

65  5.11 
!56  23.7 
57  .3.1!) 
58  2.23 

5i)  

5.31 
2^.3.13 

2.79 
3.53 

3.5.17 

2» 

2.3.43 
7.37 

5.71 
23.89 
3.7.17 
2.179 

5.7.13 

23.3.19 

2.229 
33.17 

3.5.37 
23.139 

2.33.31 
13.43 

5.131 
2«.41 
33.73 

2.7.47 

5.151   335.1) 
23.33.7:    23.107 

2.3792.3,11.13 
3.11.23     

5.191 
22.239 
3.11.29 
2.479 
7.137 

00  23.3.5 

25.5 
7.23 
2.3« 

23.41 

23.5.13 
33.29 
2.131 

23.3.11 

23.33.5 

193 

2.181 

3.113 

23.7.13 

23.5.23 

2«.5.7 
3.11.17 

2.281 

23.3.47 

23.3.5.11!    23.5.19 

23.5.43     2».3.5 
3.7.41         312 
2.431  2.13.37 
33.107 
2».33    23.241 

62  2.31 
63  35.7 
64  2« 

2.3.7.11 

2«.29 

2.331    2.3.127 
3.13.17!       7.109 
23.83;     23.191 

65  5.13 
66  2.3.11 

68  23.17 
p9  3.23 

3.5.11 

2.83 

23.3.7 

133 

5.53 

2.7.19 
3.89 
23.67 

5.73 
2.3.61 

2«°3 
33^1 

3.5.31 
2.233 

23.33.13 
7.67 

5.113 
2.283 
3«.7 
23.71 

5.7.19 
2.33.37 
23.29 
23.1(57 
3.223 

33.5.17 
2.383 
13.59 
2«.3 

5.173      5.193 
2.4332.3.7.23 
3.173     
23.7.31     23.113 
11.79.  3.17.19 

<70  2.5.7 

71 

2.5.17 
33.19 
23.43 

2.3.29 

53.7 
2«.ll 
3.59 
2.89 

2.33.5 

2.5.37 
7.53 
23.3.31 

2.11.17 

2.5.47 
3.157 
23.59 
11.43 
2.3.79 

2.3.5.19 

2.5.67 
11.61 
2=.3.7 

2.337 
33.53 

23.133 

2.3.113 

7.97 

2.5.7.11 
3.257 
23.193 

"2.33.43 

2.3.5.29 
13.67 
23.10!) 
33.97 
2.19.23 

2.5.97 

72  23.3* 

•J-O  p 

74  2.37 

2«.17 
3.7.13 
2.137 

23.11.13 
3.191 
2.7.41 

22.36 

7.139 
2.487 

75  3.5^ 
7G  23.  li) 
77  7.11 
7r  2.:i.l3 

53.11 
23.3.23 

2.139 
33.31 

3.53 
23.47 
13.29 
2.33.7 

53.19 
23.7.17 
33.53 
2.239 

53.23 
2«.33 

2.173 
3.193 

52.31 
23.97 
3.7.37 

2.389 
19.41 

53.7 
22.3.73 



2.439 
3.293 

3.52.13 

8«.61 

2.3.163' 

11.89J 

-0  2«.5 
^1  3« 
-2  2.41 

84  23.3.7 

22.335 

2.7.13 
3.61 
23.23 

2».5.7 
2.3.47 

23.71 

3.5.19 
2.11.13 
7.41 
25.32 

173 

23.519 
3.127 
2.191 

01  3 

25.3.5 
13.37 
2.241 
3.7.23 

23.112 

23.529 
7.83 
2.3.97 
11.53 
23.73 

33.5.13 
2.293 

23.3.73 
19.31 

23.5.17 
3.227 
2.11.31 

22.32.19 

23.3.5.13 
11.71 

2.17.23 
33.29 

2*.72 

2«.5.11 
2.33^73 
22.13.17 

3.5.59 
2.443 

23.3.37 
7.127 

23.5.73 
32.109 
2.491 

23.3.41 

80  5.17 
86  2.43 
87  3.29 
88  23.11 

h  — 

5.37 
2.3.31 
11.17 
23.47 
33.7 

5.7.11 

2.193 
33.43 

22.97 

2.3.5.13 
17.23 

23.73 

3.131 
2.197 

5.97 
2.3s 

23.61 
3.16^ 

5.137 
2.73 
3.22C 
2«.43 
13.53 

2.3.5.23 

22.173 
32.7.11 
2.347 

5.157 
2.3.131 

23.197 
3.203 

5.197 
2.17.29! 
3.7.47| 
23.13.19 
23.13 

bo  2.35.5 

>91  7.13 
|02  23.23 
03  3.31 
B4  2.47 

2.5.19 
~20 
2.97 
3.5.13 

23.73 

2.33.11 

2.5.29 
3.97 
23.73 

2.3.73 

5.59 
23.37 
33.11 
2.143 
13.23 

2.5.73 

23.3.41 
17.29 
2.13.19 

2.5.59 
3.197 
2«.37 

2.33.11 

2.5.79 
7.113 

23.32.]] 

1361 

2.31)7 

3.5.53 
22.199 

2.3.7.19 
17.47 

2.5.89 
3^.11 
22.223 
19.47 
2.3.149 

5.179 
2'.7 
3.13.23 
2449 
29.31 

2.33.5.11 

25.31 
3.331 
2.7.71 

flo  5.19 
|9B  2*.3 

!97  
S8  2.73 
09  3M1 

5.79 

23.33.11 

2.199 
3.7.19 

33.5.11 
2«.31 
7.71 
2.3.83 

5.7.17 
23.149 
3.199 
2.13.23 

5.139 
23.3.29 
17.41 
2.349 
3.233 

5.199 
22.3.83 

2.499 
3».37 

TABLE  OP  PRIME  AND  COrPOSITE  NUMBERS. 


379 


1 

1 

1 

r 

| 

10 

n 

12 

13 

1  ! 

15 

16 

17 

18 

19 

:J1 

22 

23 

34 

25 

26 

27 

28 

29 

3 

-J 

- 

•2. 

1! 

7 

: 
i: 

- 
3 

11 

- 

11 

2 
I 

31 

_ 

a 

': 

41 

3 
IS 

37 

• 

2 

|       5 

07    19 

C 

JT 

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; 

11 

3 

I     13 

;j 

i     - 

a 

S! 

23 

7 

3 

Ot 

• 

^ 

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- 

473 

. 

13 

j      « 

53' 

11     3 

1 

_ 

~ 

17 

3 

21 

_ 

3 

— 

3 

1 

— 



: 

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13   - 

3 

13 

3 

17 

2 

- 

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:i 

1! 

7 

a 

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29 

3 

17     3 

: 

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•r 

17 

23 

•> 

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1      3 

1 

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19    - 

:( 

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I 

17 

r. 

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7 

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i 

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M 

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3 

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3 

7 

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5 

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5 

n 

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5 

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5 

a 

3 
13 

5 

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n 

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31 

3 

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7 

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31    - 

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3 

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3 

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37 

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3 

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71 

49    - 

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19 

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43, 

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7 

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31 

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51    - 
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3 

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55     5 

3 

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57      7 

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MISCELLANEOUS   PROBLEMS.          [ART.  XXIII 


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39 

31 

17 

29     - 

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13     37 

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23 

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19 

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3.7 

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89 

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39 

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33 

93      41 

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§111.]      TABLE  OF  PRIME  AND  COMPOSITE  NUMBERS. 


383 


90 

91 

92 

93 

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95 

96 

97 

98  99 

100101 

102  103 

104  J105  106 

107   108 

109 

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101 

33 

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29       7 
11      13 

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3.5 

5   17 

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37  5.7 

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43 

13 

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5   11 

3.5 

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3.5 

07   - 

7 

11 

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23 

3 

13 

17 

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59 

11 

3   19 

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THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $ 1  • 

OVERDUE. 


j*  |       g"          /f^S\ 

MG    3  1936J 

[tiv\     1    1937 

JAN  2  01966 

;5 

REC'D  Ln 

J*N    6  '66  "^  P 

. 

- 

LD  21-100m-8,'34 


"YB   17400 


614/02, 
M3 


11188: 


